Can continuous functions be made smooth by changing the smooth structure on the domain?

Usually not. Let's take $M = \mathbb{R}$ and let $f \colon \mathbb{R} \rightarrow \mathbb{R}$ be some continuous nowhere differentiable function. If $M$ has an atlas under which $f$ becomes a smooth map then for any $p \in \mathbb{R}$ you can find a homeomorphism $\phi \colon I \rightarrow U_p$ where $I \subseteq \mathbb{R}$ is an open interval and $U_p$ is an open interval around $p$ such that $f|_{U_p} \circ \phi = g \colon I \rightarrow \mathbb{R}$ is smooth. But then $f|_{U_p} = g \circ \phi^{-1}$ is the composition of a smooth map and a homeomorphism between open intervals. But any such homeomorphism must be monotone and any monotone function is differentiable almost everywhere which then implies that $f|_{U_p}$ must be differentiable almost everywhere, a contradiction.