How could Tycho Brahe determine positions without accurate clocks? Tycho Brahe determined the positions of stars and planets to an accuracy of 2 minutes of angle. Pendulum clocks hadn't been invented yet so he couldn't have known the time to better than 15 minutes. Wouldn't he need a more accurate clock to measure celestial positions?
 A: Most already said in other answers, But in his Tycho Brahe: a picture of scientific life and work in the sixteenth century, John Louis Emil Dreyer writes about how time was determined by Tycho and with some historical context which I find important. The relevant text is attached at the end.
It also worth noting that Tycho himself was not shy to record timing of observation sometimes to the resolution of mere 10 seconds of time. (though most of his observation are in whole minutes). Here is a random example from one of his books of observations records. (this is observation of Mars from 4 March 1598 [Not sure if Gerogian calender]) where we see times in 1/6 of the minute.:


*

*It is right, of course, that to find a current location of a celestial object the time itself in unnecessary, but time of the observation was import when building the model of the movement. What resolution was needed? If we look at the fastest moving object (versus the fixed) which is the Moon. The Moon moves more than 12deg is 24h, which is 30s of arc in 1m of time. Hence it was important for Tycho to know the time in the resultion of at least 1 minute.


*PM 2Ring rightly notes that "Tycho didn't use right ascension and declination for his coordinates, he used ecliptic longitude and latitude." But it should be clarified that the ecliptic coordinate system was indeed used for the planets models, but in the observations themselves Tycho usually (but not always) found the location of the planets in equatorial coordinate system and then it was converted if this observation was needed.

here is the Text from Tycho Brahe: a picture of scientific life and work in the sixteenth century, John Louis Emil Dreyer:

An important use to which the quadrants were put at
Uraniborg was the determination of time. At Alexandria the
beginning, middle, or end of the hour was generally the
only indication of time which accompanied the observations
of planets, which was perhaps sufficient, owing to the
limited accuracy of the observations. The time was found
by water- or sand-clocks, which were verified by observing
the culmination of some of the forty-four stars which
Hipparchus had selected so well that the time could be
determined with an error not much exceeding a minute.[18]
An important step forward as regards the accurate
determination of time was made by the Arabs in the ninth
century. Ibn Yunis mentions a solar eclipse observed at
Bagdad on the 30th November 829 by Ahmed Ibn
Abdallah, called Habash, who at the beginning of the
eclipse found the altitude of the sun to be 7°, while at the
end the altitude was 24°. This seems to have been the
earliest though crude attempt to use observations of altitude
to indicate time, but the advantage of the method was
evident, and at the lunar eclipse on the 12th August 854 the
altitude of Aldebaran was measured equal to 45° 30'. Ibn
Yunis adds that he from this made out the hour-angle to be
44° by means of a planisphere. Ibn Yunis communicates a
number of other instances from the tenth century,[19] but the
instruments used were very small, and only divided into
degrees; and though Al Battani gave formulæ for the
computation of the hour-angle, the Arabians generally
contented themselves with the approximate graphical
determination by the so-called astrolabe or planisphere.


In Europe the use of observations of altitude for
determining time was introduced in 1457 by Purbach, who,
at the beginning and end of the lunar eclipse on the 3rd
450
September, measured the altitude of "penultima ex
Plejadibus."[20] Bernhard Walther was the first to introduce
in observatoriesthe use of clocks driven by weights. Thus
we find among his observations one of the rising of
Mercury. At the time of rising he attached the weight to a
clock of which the hour-wheel had fifty-six teeth, and as
one hour and thirty-five teeth passed before the sun rose, he
concluded that the interval had been one hour thirty-seven
minutes. Walther adds that this clock was a very good one,
and indicated correctly the interval between two successive
noons; but all the same he must have seen how unreliable it
was, for though he used the clock during the lunar eclipse in
1487, he at the same time measured some altitudes.[21]


In Tycho Brahe's observatory the clocks never played an
important part. Though he possessed three or four clocks,
he does not anywhere describe them in detail, while he in
several places remarks that he did not depend on them, as
their rate varied considerably even during short intervals,
which he attributed to atmospherical changes (although he
kept them in heated rooms in winter), as well as to
imperfections in the wheels. At the side of the mural
quadrant he had placed two clocks, indicating both minutes
and seconds, in order that one might control the other, and
in the southern observatory was a large clock (horologium
majus) with all the wheels of brass. Whether Bürgi, during
Tycho's residence at Prague, supplied him with a pendulum
clock, as stated by a later writer,[22] must remain very
doubtful, but that Tycho did not possess such a clock at
Uraniborg seems certain, as he would not have neglected to
describe so important an addition to his stock of
instruments. As he found the clocks so uncertain, Tycho
also tried time-keepers similar to the clepsydræ of the
ancients, which measured time by a quantity of mercury
flowing out through a small hole in the bottom of a vessel,
in which the mercury was kept at a constant height, in order
that the outflow might not vary with the varying weight of
the mercury. By ascertaining the quantity of mercury which
flowed out in twenty-four hours, it was easy to make out the
interval which passed between the culmination of the sun
and a star by starting the time-keeper when the former
passed the meridian, and letting it run until the latter passed,
and then weighing the amount which had flowed out.
Instead of mercury, Tycho also tried lead monoxide powder,
and adds to his account of these experiments some remarks
about Mercury and Saturn (lead), and their astrological
relations, which naturally suggested themselves to his mind.
[23] But he does not seem to have used these clepsydræ
except by way of experiment, and his methods of observing
made him in most cases independent both of them and of
the clocks. In addition to the altitudes (about which he
justly remarks that they must not be taken too near the
meridian, where they vary very slowly, nor near the
horizon, where they are much affected by refraction), he
observed hour-angles of the sun or standard stars with the
armillæ to control the indications of his clocks, and his
observations of the moon, comets, eclipses, &c., where
accurate time determinations are indispensable, were
thereby doubly valuable. Occasionally azimuths were also
observed for the same purpose, the zero of the azimuth
circle having been found by observing the east and west
elongation of the Pole Star.

A: In observational astronomy the position of objects in the sky
(like stars and planets) is given in the equatorial coordinate
system by two angles: declination and right ascension.
In this coordinate system stars are fixed,
and planets move very slowly with respect to the stars.
Using this coordinate system has the advantage that the coordinates
of astronomical objects do not depend on the local time and the
local position of the observer on earth.
So there is no need for an accurate clock, or a clock at all.
The image below shows the path of Mars in 2022 relative to the
fixed background of stars.
The vertical scale is declination, and the horizontal scale
is right ascension (traditionally given in a $24$ h scale,
instead of a $360°$ scale).
Notice that here the speed of Mars is not larger than $0.5°/$day.

(image from The position of Mars in the night sky)
A: The answer by PM-2Ring makes an excellent and often overlooked point: before clocks, astronomers were not dependent on crude mechanical devices, they had star-based timekeeping.
It is worth noting also that some ancient star-based methods were fairly simple in application, they used observed altitudes of the sun and/or stars, along with tables relating these altitudes to date and time. See for example Ibn Yunus' Very Useful Tables (for reckoning time by the sun), described by D A King (1973), in Archives for History of Exact Sciences vol. 10, pp. 342-394. In the ancient and medieval world, precise timekeeping was culturally important e.g. in Muslim religious practice, hence a profusion of many quite sophisticated methods.
As for the way that Tycho Brahe applied time-keeping methods, it is a pity that many descriptive references to him say nothing about his time-keeping practices at all. But the biography 'Tycho Brahe' by J L E Dreyer (1890) gives a number of items of information about this, and some quotations may be of use here as the book may not be very easily accessible.
About Tycho's early years, especially 1577, the year of the comet, Dreyer noted:

"When observing this comet, Tycho had not yet at
his disposal as many instruments and observers as
in after-years, nor had he as yet perceived the
necessity of accurate daily time determinations by
observing altitudes of stars, but [he] merely
corrected his clocks by sunset." (p.159.)

In Tycho's established practice after building his installations at Hven, according to Dreyer (p.322-):

"An important use to which the quadrants were put at
Uraniborg was the determination of time"

(through the taking of altitudes).
These quadrants were of the kind known as  "quadrans azimuthalis," of which Tycho constructed four. They

"were extensively used ... chiefly for merely
observing altitudes, while the azimuths"

(a measurement for which they could also be used)

"were rarely taken, especially during his later
years. The largest quadrant {quadrans magnus
chalibetis) was enclosed in a square (also of
steel), of which the side was equal to the radius
of the quadrant. Two of the sides were graduated,
and the alidade pointed to these graduations as
well as to those on the arc, so that the instrument
was a combination of a quadrant and the "quadratum
geometricum" of Purbach (which the Arabians had
also known), which increased the solidity of the
instrument."


"In Tycho Brahe's observatory the docks never played
an important part. Though he possessed three or four
clocks, he does not anywhere describe them in detail,
while he in several places remarks that he did not
depend on them, as their rate varied considerably even
during short intervals, which he attributed to
atmospherical changes (although he kept them in heated
rooms in winter), as well as to imperfections in the
wheels. At the side of the mural quadrant he had placed
two clocks, indicating both minutes and seconds, in
order that one might control the other, and in the
southern observatory was a large clock (Horologium
majus) with all the wheels of brass."

Dreyer also notes (p.325) that

Tycho's "methods of observing made him in most cases
independent both of" (equipments such as clepsydrae)
"and of the clocks. In addition to the altitudes (about
which he justly remarks that they must not be taken
too near the meridian, where they vary very slowly, nor
near the horizon, where they are much affected by
refraction), he observed hour-angles of the sun or
standard stars with the armillae to control the
indications of his clocks, and his observations of the
moon, comets, eclipses, &c, where accurate time-
determinations are indispensable, were thereby doubly
valuable. Occasionally azimuths were also observed for
the same purpose, the zero of the azimuth circle having
been found by observing the east and west elongation
of the Pole Star."


"For observations of altitude Tycho also used a
sextant of 5 1/2 feet radius, turning on a vertical
axis, with one end-radius kept horizontal by means
of a plumb-line attached to the centre of the radius."

Dreyer also described some of the arrangements for accurately dividing  the scales of quadrants and sextants, and the heritage
of methods such as those of the Arabian astronomers e.g. for
translating altitudes to times -- already mentioned above.
Tycho suffered disruption of his observing practices of the
established years at Hven  when he moved to Prague in 1597-8.
Dreyer's biography comments (p.258):

"By the beginning of February 1598 ... [in Prague] ...
he was again able to use quadrants for determining the
time by altitude observations."

More information about Tycho's timekeeping is available
in J L E Dreyer's multi-volume edition of Tycho's works and
observations published at Copenhagen in the early part of the
20th century.
So it does seem clear that in spite of the scarcity now of relevant descriptions, there is enough historical material to show that Tycho had at his disposal methods with an established history and of some sophistication: He was not left subject to the vagaries of clocks and other such crude devices of the time.
A: As an aside- Tycho achieved his legendary accuracy by building ever-larger angle measurement devices called quadrants. The largest was about 6 feet in diameter and he aimed its sighting line at a star by placing his eye next to a sight at one end of the sighting line and rotating the apparatus until the other sighting point on the opposite end of the sighting line was zeroed on the star. He was the last astronomer to measure stellar positions without an optical telescope.
A: Tycho Brahe had access to an excellent clock: the sky itself!
As others have mentioned, astronomers have several coordinate systems for specifying the positions of bodies on the celestial sphere. Because the stars are so far away their positions on the celestial sphere are (almost) constant. They do move slightly, but you need good telescopic observation to detect that. Traditionally, the stars were often referred to as the "fixed stars".
The rotation of the celestial sphere is, of course, due to the rotation of the Earth on its axis. That rotation is quite regular, so we can use it as a clock. If you know the Right Ascension of a visible star, you can determine the current sidereal time by measuring the star's angular distance from the meridian.
Tycho's primary mission was to improve the accuracy of the known star coordinates. He (and his workers) made hundreds of great star position measurements. (They weren't quite as good as Tycho believed, though, and were superseded in subsequent decades by telescopic observations). Tycho didn't use right ascension and declination for his coordinates, he used ecliptic longitude and latitude.
Tycho also observed the Moon and the planets, which move relative to the fixed stars. You don't need to worry about time when making a star map, but it is important to include the time when recording the positions of the Moon and planets. Tycho could certainly have used sidereal time for that purpose, he may have even had some crude mechanical clocks that were regularly set using sidereal time measurements. In ancient eras, it was common to specify the time of an astronomical phenomenon by stating how far some prominent star was from the meridian. The Babylonian astronomers used that system.
Famously, Johannes Kepler developed his three laws of planetary motion using Tycho's data for Mars. (Kepler worked extensively with Tycho, and made numerous observations, but Tycho owned the data).

Sidereal time is a very uniform time scale. It wasn't until the 20th century that we had mechanical clocks that were precise enough to measure the variations due to the slight irregularities in the Earth's rotation.
Solar time (i.e., time measured by the position of the Sun) isn't quite so regular. That's because of the tilt of the Earth's axis, and the eccentricity of its orbit, as I explain here. Here's a graph showing the approximate difference between mean and apparent solar times. This difference is known as the Equation of Time.

Until the development of precise mechanical clocks, most people used some form of local solar time, as shown on a sundial. People who needed a uniform time scale (mostly astronomers) used sidereal time. Mean solar time was invented because it was too hard for early clocks to track apparent solar time. Precision sundials that incorporate the Equation of Time, known as heliochronometers, were often used to set mechanical clocks; the French railways were still using heliochronometers in the early years of the 20th century.
For most of recorded history, the motion of the celestial bodies was the basis of all precision timekeeping. In the late 1960s, we switched over to atomic time as the basis of our timekeeping systems, but astronomy still has some input in the determination of Universal Time because we want the days measured by our clocks and calendars to remain related to the Sun.
Modern high-precision timekeeping is rather complicated. You can read about it in the articles by Steve Allen on the Lick Observatory site: A Brief History of Time Scales. Precise data regarding the Earth's rotation is gathered and maintained by the IERS, the International Earth Rotation Service.
A: Promoted from a comment:

No, in astronomy the position of planets are given relative to the stars (by giving right ascension and declination). Planets move very slowly relative to the stars (at most a few degrees per day). Thomas Fritsch

