"right" uniformity condition for Nick's Class
You can use $\mathsf{DLogTime}$ for uniformity of $\mathsf{NC}$ and $\mathsf{NC^2}$. There is no problem and the classes uniform $\mathsf{NC^k}$ remain the same and equal to $\mathsf{ATimeSpace}(O(\lg^k n),O(\lg n))$ (for $k\geq1$).

Generally, the only case that we need to be more careful is $\mathsf{NC^1}$ case where one should be careful about what needs to be in decidable in $\mathsf{DLogTime}$. If you use extended connection language description of circuits then everything works even in $\mathsf{NC^1}$ case.

For more on uniformity see:

Walter L. Ruzzo, "On Uniform Circuit Complexity", Journal of Computer and System Sciences, vol. 22 (1981), pp. 365–383.