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.
The nerve functor $N : \mathrm{sSet} \to \mathrm{Spc}(k)$ 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 $|-|_k : \mathrm{Spc}(k) \to \mathrm{sSet}$. 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 $$\mathrm{Sing}_*(K(\mathbf{Z}(n), 2n)) : \mathrm{Sm}(k) \to \mathrm{sSet} $$ 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.