First an unimportant detail: Ole Rømer described the assessment of the speed of light that he is rightly remembered for,
but to my knowledge he wasn't actually in the position to obtain measurements that were accurate enough.
Correction in response to a comment: I misremembered.
Rømer's ideas and results were known to his contemporaries through correspondence. It was just that these ideas were not formally published.
Before I proceed there is a essential point I need to discuss. Some people regard the question of whether a one-way measurement of the speed of light is possible in connection with questioning whether it is possible to set up an experimentum crucis to distinguish between a Lorentz aether theory on one hand or special relativit on the other hand. My expectection is that no such experiment is possible. That is: I expect any one-way measurement of the velocity of light to find the same value as a two-way measurement.
If it is granted that we expect the moons of other planets to move according to the the laws of motion then the shift in the timing can be attributed to different time transit delay depending on travel distance. We know the distances between the planets in the solar system. Putting al that together the speed of light can be inferred.
The only criticism one can level against that is that it is a process with several stages of inferring.
Then again, multiple stages of inferring is how the majority of measurements are obtained. (Example: distances to other galaxies. Distances to nearby Cepheids are assessed by parallax measurement. Distant Cepheids are identifed and the distance is inferred from the measured luminosity.)
A two-way measurement of the speed of light requires far less stages of inferring, if any.
The intellectual challenge is the issue whether it is possible to perform a one-way measurement of the speed of light with a setup that is just as simple as a setup for a two-way measurement of light.
About time-keeping on Earth in general:
There are multiple centers for time-keeping around the world, and they work together to provide the world with a coordinated world time.
Having a high precision coordinated world time is crucially important for astronomy. There are forms of radio astronomy where each observatory records data with time labeling at the highest possible level. Disk drives containing these data are from over the world shipped to a central facility. In this facility the data are processed, and because of the high precision of time labeling they can correlate the signal phase. Thus data from distant observatories can be processed as if those observatories are at oppposite ends of the perimeter of a planet sized radio receiver dish.
Now comes the important part:
The way world coordinated time is maintained is not by Einstein synchronization. For this specific application Einstein synchronization would not work.
To explain why it wouldn't work I will describe a highly stylized version of how world time coordination is done; the underlying principle is the same.
Image a number of time centers located at equal intervals along the equator. Let me make that (arbitrary choice) 12 centers.
To maintain synchronized time they are continuously relaying time signals, both in clockwise direction and in counterclockwise direction.
Let me say that 12 signals are relaying in each directions. Each signal is a pulse, with an identifyer.
The procedure is that on each relay step there is an option to make a minimal adjustment such that the time spacing between the 12 clockwise propagating pulses is as close to equal as possible, and that the time between the 12 counterclockwise propagatign pulses is as close to equal as possible.
If the Earth would not be rotating then the counter-propatating trains of pulses would behave the same.
On our rotating Earth the rotation introduces a bias. The pulses traveling from west to east take a bit longer to make it from station to station. Conversely, the pulses traveling from east to west take less time.
The animation depicts the phenomenon. In interferometry this phenomenon is called 'the Sagnac effect'. One particular form of interferometery, ring laser interferometry, uses the Sagnac effect to measure the Earth rotation rate (and minute variations of it).
The underlying principle of the Sagnac effect is that the speed of light is the same in all directions.
The blue and red dots represent counterpropagating pulses. The counterpropagating pulses have the same speed everywhere. Note that in the animation they cross each other at the same spot all the time, because they are propagating at the same speed.
The four grey dots represent relay stations along the way.
The operators of the relay stations know the rotation rate of the system. The operators know how much time interval to expect between the blue pulses and the red pulses. Even if the operators would not know in advance what the rotation rate of the system is, they can infer that rotation rate from the difference in travel time between the blue and red pulses.
What would happen with Einstein synchronization
Applying Einstein synchronization would fail to meet the demands:
The 12 stations would have to pair up. 1-2, 2-3, 3-4 ... 11-12 etc. They can do all those pair-wise synchronizations, with application of Einstein synchronisation. Except: then you cannot close the loop. If all pairs have applied Einstein synchronization then for the pair 12-1 there will be a time gap.
For world coordinated time such a gap is unacceptable. The procedure to maintain world coordinated time must take the Earth's rotation into account, so that is what happens.
On the real Earth the time keeping centers are not equally spaced of course, so that has to be taken into account. Also, they are at different elevation above sea level, hence at different geopotential height which gives rise to difference in gravitational time dilation effect, which has to be taken into account. Clearly there are a lot of difficulties; evidently the engineers have overcome all of them.
The precise time-keeping that enables worldwide consistent GPS is dependent on having a single coordinated world time.
Also, given that a single coordinated world time is used, and given that the Earth is rotating: depending on where a GPS satellite is relative to the GPS receiver the signal may be traveling east-to-west or west-to-east. For west-to-east travel you expect more travel time than for east-to-west travel. The GPS technology takes this difference into account.