Variation of Ascoli-Arzelà theorem for $C^1$ functions

Indeed, the idea of approximating $\Omega$ by a sequence of compact sets will work. Let $$ K_j:=\left\{x\in\mathbb R^n, \lVert x\rVert\leqslant j\right\}\cap \left\{x\in\mathbb R^n, d\left(x,\mathbb R^n\setminus\Omega \right)\leqslant 1/j\right\}, $$ where $d(x,S)=\inf\{\lVert x-y\rVert,y\in S\}$ and $\lVert \cdot\rVert$ denotes the Euclidean norm. Then $K_j$ is compact for each $j$ and $\Omega=\bigcup_{j\geqslant 1}K_j$. Moreover, notice that each compact subset of $\Omega$ is contained in some $\Omega_j$.

Now you can use a diagonal extraction process to find a subsequence for which the uniform convergence holds for each $K_j$ and in view of the previous remark, for each compact subset of $\Omega$.