Are circularly defined {velocity, distance, and time} a problem in physics? In order to measure velocity, one needs a calibrated measuring stick and clock. But in order to calibrate a measuring stick you need a calibrated clock and velocity. And in order to calibrate a clock you need a calibrated measuring stick and velocity. Thus velocity, distance, and time are all defined in terms of each other. Is this circularity a problem that needs to be addressed by a fundamental theory of the universe? And is it addressed by any theories on the market?
 A: No, and no. For several reasons:


*

*You do not need a calibrated measuring stick and clock to measure distance and time. You can pick up a random piece of wood and call that your length unit, and measure time using any convenient repetitive physical process, such as your pulse. (Of course a pulse does not make a particularly accurate clock, but just pretend for a minute.) The issue of calibration only arises when you need to make different measuring devices agree with each other.

*If you do need to calibrate, say, a measuring stick, the easiest way is to just compare it with another measuring stick. Or to calibrate a clock, you compare it with another clock.  You do not need a calibrated measuring stick and velocity. There are many extremely reliable natural clocks that can be used for this purpose. For example, our current time standard is defined in terms of the oscillation period of the EM radiation emitted by a hyperfine transition of the cesium atom. As long as you're sufficiently careful when you measure it, this provides an easily available, naturally pre-calibrated clock.

*We also have a natural pre-calibrated velocity, namely the speed of light. Again, as long as you're sufficiently careful when you measure it, this provides an easily available, naturally pre-calibrated velocity. In combination with the definition of the time scale using the cesium HF transition, we use this to define our unit of length.

A: You are getting confused because you are reading somewhere that the speed of light can meaninfully vary. Then if the speed of light changes, what does it change relative to? How can we tell if the speed of light just doubled, or if all the atoms in the universe suddenly shrunk by a factor of 2? Obviously you can't. There is no difference between these two statements, even philsophically.
There is no meaning to the statement that dimensional constants change. You are being confused by incompetent physicists that claim otherwise, because they want to sensationalize to the general lay public, which does not appreciate this well established fact. You asked this question because you realize that it is meaningless to speak about a dimensional constant changing, and this is causing cognitive dissonance. All physicists go through this cognitive dissonance, and it is fixed when you understand that dimensional constants cannot be meaningfully said to vary
A: In a loose sense (you need to add a mass constant, to speak of momentum instead of velocity), the circularity is addressed by the three constants $\hbar, G, c$ and their respective theories, quantum mechanics, general relativity, and special relativity.
As say by David, $c$ solves the space vs time. 
For speed vs distance, it is $\hbar$ what is needed, and in fact the lack of this constant delayed the publication of Newton's Principia, who -after suggestions of two friends- did a lot of rewritting for the theorem of angular momentum, aka equal areas at equal times. Note that the area in an orbit, at fixed $\delta t$, is proportional to speed cross times distance.
I can not argue so easily for speed against time, but if it can be reduced to an issue of spacetime geometry, then $G$ controls it, but again there is the issue that you have no considered mass in your circles. Here it could be more about acceleration or about force.
Another way to see all of this is that in natural units $\hbar=1, c=1$ everything can be written in energy, say $eV$, and then $G$ provides you a natural calibrated stick, Planck mass.
So the second point of your question is yes, it is addressed.
As for the first question, ¿is the circularity a problem for a fundamental theory?, amazingly it seems not, as it is proved by the fact that Newton work was seen as a valid fundamental theory for centuries. 
A: My answer PSE-here is part of my answer. From there:
"length, time, mass and charge units are deeply linked through the properties of atoms and speed of light." 
Taking $c,\varepsilon,G$ as constants ("field constants" because they depend on space properties) the bulk constitutive equations of the world keep good as long as  Mass, Charge, Length, Time units  (M-L-Q-T) change by the same scale factor.  c speed is an invariant (L/T) and it is imposed by space.
Double your atom's mass/size/time/charge and the world will behave as usual.
There are some people that (implicit or explicitly)  ascribe a special character of absoluteness to the size of atoms around us.  I'm sure that they can not offer a plausible reason to that beleif.
It is the relation [c]=[L]/[T]  that allow us to untie the circular link, i.e. the units L and T are not independent.  
