$\newcommand\C{\mathcal C}$If $\C$ is convex, then the projection onto the convex set $X\C$ is uniquely defined and $1$-Lipschitz, so that 
$\|X\hat w-Xw^\ast\|\le\|v-Ev\|$ and hence $\|\hat w-w^\ast\|\le\|X^{-1}\|\|v-Ev\|$. So, for instance,
$$P(\|\hat w-w^\ast\|\ge t)\le\|X^{-1}\|^2\frac{E\|v-Ev\|^2}{t^2}$$
for real $t>0$. 

If $\C$ is not convex, then the projection onto the convex set $X\C$ is in general not uniquely defined and is discontinuous, so that in that case no concentration result is possible.