One way to understand your question is in the framework of $\mathbf{A}^1$-homotopy theory. This is because your nerve functor is better understood when defined on a cocomplete category like the category of motivic spaces $\mathrm{Spc}(k)$. The latter is roughly speaking the free $\infty$-cocompletion of the category $\mathrm{Sm}(k)$ of smooth $k$-schemes, localized with respect to Nisnevich covers and $\mathbf{A}^1$-equivalences. (So its objects are $\infty$-presheaves, and it is presented by a model structure on the category of simplicial presheaves on $\mathrm{Sm}(k)$.)

The nerve functor $N : \mathrm{Spc}(k) \to \mathrm{sSet}$ is just the restriction of scalars functor induced from $\Delta^\bullet_k : \mathbf{\Delta} \to \mathrm{Spc}(k)$. Since $\mathrm{Spc}(k)$ is cocomplete, by abstract nonsense (as tetrapharmakon mentioned in his answer) this admits a left adjoint which is the (motivic) geometric realization functor $|-|_k : \mathrm{sSet} \to \mathrm{Spc}(k)$. Certainly these constructions have been studied in this setting, in various papers of Voevodsky for example, though I am not sure if he ever used the term "nerve".

There is another functor $\mathrm{Sing}_* : \mathrm{Spc}_k \to \mathrm{Spc}_k$ which is given by the formula
$$ \mathrm{Sing}_n(X)(U) = \mathrm{Hom}_{\mathrm{Spc}(k)}(U \times \Delta^n_k, X) $$
for a smooth scheme $U$. This seems to be a more refined version of the construction of Suslin-Voevodsky referenced by Tom Goodwillie, and I believe that these simplicial sets are more directly analogous to the nerve or singular simplicial complex in topology than the above nerve $N$. When one takes for $X$ the motivic Eilenberg-Mac Lane spaces, the induced functor
$$\mathrm{Sing}_*(K(\mathbf{Z}(n), 2n)) : \mathrm{Sm}(k)^{\mathrm{op}} \to \mathrm{sSet} $$
is particularly interesting.
For a "good" smooth scheme $U$, the simplicial set it gives has its homotopy groups identified with the motivic cohomology of $U$. So certainly the topology of this simplicial set is relevant to the study of cohomological invariants.

Surely there are many more interesting things to be said, but I am not an expert so I will refer to papers of Voevodsky. His ICM talk may be a good place to start.

**Edit:** The fact that $\mathrm{Sing}_*$ should be viewed as a refined version of the nerve $N : \mathrm{sSet} \to \mathrm{Spc}(k)$ can be made more precise by the following observation: $\mathrm{Spc}(k)$ is presented by a simplicial model category, and hence is tensored and powered over $\mathrm{sSet}$. This means in particular that there are adjoint pairs $(- \otimes \Delta^n, \underline{\mathrm{Hom}}(\Delta^n, -))$ for each $n$. For $X \in \mathrm{Spc}(k)$ each $\underline{\mathrm{Hom}}(\Delta^n, X)$ is a actually a presheaf (and not a set, like $N(X)_n = \mathrm{Hom}(\Delta^n, X)$). And the point is that $\mathrm{Sing}_n$ is by definition precisely the power functor $\underline{\mathrm{Hom}}(\Delta^n, -)$.

**Edit 2:** Since you asked for a reference, I just noticed that there is a good discussion about the functor $\mathrm{Sing}_*$ in section 2.3.2 of

- Fabien Morel, Vladimir Voevodsky,
*A^1-homotopy theory of schemes*, 1998, web.