Let's say that one has two varieties $V_1, V_2$, with singularities at points $x_1$ and $x_2$ respectively, and that there exist open neighbourhoods $U_1$ and $U_2$ of these singularities such that $U_1$ and $U_2$ are isomorphic, i.e. such that $V_1$ and $V_2$ are locally isomorphic around their singularities.
Is it then always a given that if we have minimal resolutions $W_1$ and $W_2$ of $V_1$ and $V_2$ at these singularities, then around the exceptional divisors $D_1$ and $D_2$, there exists open neighbourhoods $U'_1$ and $U'_2$ such that $U'_1$ and $U'_2$ are isomorphic?
Further, if this is the case, is it then a given that $\mathcal{O}_{W_1} (U'_1) \cong \mathcal{O}_{W_2} (U'_2)$?