Are Chow groups invariant under universal homeomorphisms?

If you work in equal characteristic, you have a chance of survival. In characteristic zero, this amounts to check that Chow groups do see nilpotent extensions. In characteristic $p>0$, if $p$ is invertible in your coeffcient Ring $R$, what you want is true as well (in fact this is true for higher Chow groups). I do not know any explicit reference though (but this does not mean such thing does not exist). However, this is an easy consequence of combinations of results which are documented.

To see it first observe that higher Chow groups are representable in $DM^{eff}(k,R)$ at least for equidimensional affine schemes: this is Theorem 4.2 in Bivariant cycle cohomology, by Friedlander and Voevodsky (in the book "Triangulated categories of motives"). This extends to non affine schemes as follows: using Suslin's beautiful paper Motivic complexes over nonperfect fields, we may replace $k$ by its perfection without changing $DM^{eff}$ with $R$ coefficients, and then apply the main result from Kelly's book which allows to replace arguments of resolution singularities as in the paper of Friedlander and Voevodsky above by resolution by $\ell$-alterations with $\ell\neq p$, which exist by a well known result of Gabber.

Finally, note that all motivic invariants tend to be invariant under universal homeomorphisms after you invert the exponential characteristic: this may be seen a joint paper of mine with Déglise for $DM$ (a way to summarize what I wrote above is that, using Suslin's paper, alle the representability results in section 8 of my joint paper with Déglise are true over arbitrary fields), and, more generally, in a paper of Khan and Elmanto for $SH$.

If you are happy to tensor with $\mathbf{Q}$, then all this extend to arbitrary schemes, since $DM$ satisfies $h$-descent with rational coefficients.