The following is stated in the paper
Moshe Marcus and Victor J. Mizel, MR 531975 Complete characterization of functions which act, via superposition, on Sobolev spaces, Trans. Amer. Math. Soc. 251 (1979), 187--218.
as Theorem 1 (note that $T_f$ is the superposition operator corresponding to $f$ and that the paper assumes $\Omega$ to satisfy the cone condition):
Suppose that $\Omega$ is bounded. Let $f \colon \mathbb R^m \to \mathbb R$ be a Borel function and let $p$, $r$ be two such that $1 \le r \le p < N$. Then $T_f$ maps $W^{1,p}(\Omega)^m$ into $W^{1,r}(\Omega)$ if and only if the following conditions hold:
- $f$ is locally Lipschitz in $\mathbb R^m$
- The first order partial derivatives of $f$ satisfy the inequality $$ \left| \frac {\partial f}{\partial \xi_i}(\xi) \right| \le a_0(1+|\xi|^\nu)\ \text{a.e. in $\mathbb R^m$, $i = 1,\dotsc,m$}$$ where $a_0$ is a constant and $\nu = N(p-r)/(r(N-p))$.
If $N < p$ (or $N = 1$ and $1 \le p$) and $1 \le r \le p$ then $T_f$ maps $W^{1,p}(\Omega)^m$ into $W^{1,r}(\Omega)$ if and only if condition 1 holds.
The theorem goes on to state that under these circumstances $T_f$ is a continuous operator; bounds are established for its image, too. The case of unbounded $\Omega$ is considered in the same paper, see Theorem 3.