It seems natural to look at, even with $G$ non-reductive.

For instance, let $G=G_m$ be a $2m+1$-dimensional Heisenberg group: I write $G=V\times K$, where $(V,\phi)$ is a $2m$-dimensional symplectic space over $K$ (not of characteristic $2$) and law $(X,T)\cdot (Y,U)=(X+Y,T+U+\phi(X,Y))$. In particular (with suitable sign conventions), the group commutator $[(X,T),(Y,U)]$ is $(0,2\phi(X,Y))$.

Consider an $2n$-tuple in $G^n$, written as $((x,t),(y,u))$ with $(x,t),(y,u)\in G^n$ (so $x,y\in V^n$ and $t,u\in K^n$), $x=(x_i)_{1\le i\le n}$, $y=(y_i)_{1\le i\le n}$, with $x_i,y_i\in V$, and $T=(t_i)_{1\le i\le n}$, $U=(u_i)_{1\le i\le n}$, with $t_i,u_i\in K$.

The condition $\prod_{i=1}^n[(x_i,t_i),(y_i,u_i)]=e$ can be written as $\sum_i\phi(x_i,y_i)=0$. That is, $\Phi(x,y)=0$, where $\Phi(x,y)$ is defined as $\sum_i\phi(x_i,y_i)$, which makes $(V^n,\Phi)$ a $2mn$-dimensional symplectic space. Note that if we ignore the $t_i,u_i$, we obtain the hypersurface $$H_{mn}=\{(x,y)\in V^{2n}:\Phi(x,y)=0\}$$ in the $4mn$-dimensional space $V^{2n}$, which is non-singular outside zero (and the $G$-action by conjugation is trivial).

So if we consider the space $\mathrm{Hom}(\Gamma_n,G_m)$ (where $\Gamma_n$ is the genus $n\ge 1$ surface group), we have its description as $H_{mn}\times K^{2n}$, and it's non-singular outside $\{0\}\times K^{2n}$.

The conjugation action (after renormalization by $2$ and factoring the action through $V$) consists in letting $V$ act on this product as
$$z\cdot (x,y,t,u)=(x,y,t+\psi(z,x),u+\psi(z,y)).$$
Here $\psi(z,x)$, for $x\in V^n$ and $z\in V$, is defined as $(\phi(z,x_i))_{1\le i\le n}$.

Since for $(x,y)\neq (0,0)$, the linear map $z\mapsto\psi(z,x),u+\psi(z,y)$ is injective, it's not hard to prove that the quotient remains smooth outside $\bar{p}^{-1}(\{(0,0)\})$, where $p$ is the projection $\mathrm{Hom}(\Gamma_n,G_m)\to H_{mn}$ and factors through a projection $\bar{p}:\mathrm{Hom}(\Gamma_n,G_m)/G_m\to H_{mn}$.

Note that fibers of $p$ are "translates" of $K^{2n}$; each fiber of $\bar{p}$ naturally carries a structure of affine space of dimension $2n-1$, except the fiber of $(0,0)$ which is still naturally identified to $K^{2n}$.

On generalizations:

Let $G$ be an arbitrary $s$-step nilpotent simply connected Lie group (that is, $G^{s+1}=\{1\}$ where $(G^i)_{i\ge 1}$ is the lower central series), and $\Gamma$ is an arbitrary finitely generated group. Let $\Gamma(s)$ be the real Malcev completion of the nilpotent quotient $\Gamma/\Gamma^{s+1}$ (this is a a simply connected $s$-step nilpotent Lie group in which $\Gamma/\Gamma^{s+1}$ modulo its finite torsion subgroup, sits as a lattice. Then we can identify $\mathrm{Hom}(\Gamma,G)$ to $\mathrm{Hom}_{\mathbf{TopGrp}}(\Gamma(s),G)$, and this identification commutes with the $G$-action. Furthermore, letting $\Gamma[s]$ be the Lie algebra of $\Gamma(s)$ and $\mathfrak{g}$ that of $G$, this can be identified to $\mathrm{Hom}_{\mathbf{R}\text{-}\mathbf{LieAlg}}(\Gamma[s],\mathfrak{g})$.

Hence all the study with nilpotent target reduces to that of spaces of homomorphisms between Lie algebras.