Let $R=k[x_1,..,x_n]/I$ and let $X=Spec(R)$ be it's associated affine scheme. Suppose that $X$ has only one isolated singularity, say at the origin $\mathfrak{m}=\langle x_1,...,x_n\rangle$. Now, let $R_{\mathfrak{m}}$ be the localization at $\mathfrak{m}$, which is a local ring with maximal ideal $\mathfrak{m}_{\mathfrak{m}}$ that is not regular based on the assumption that $X$ is singular at $\mathfrak{m}$. Take $Bl_{\mathfrak{m}_{\mathfrak{m}}}(Spec(R_{\mathfrak{m}}))=Proj(R_{\mathfrak{m}}[\mathfrak{m}_\mathfrak{m}t])$ to be the blowup of $Spec(R_{\mathfrak{m}})$ at the point $\mathfrak{m}_{\mathfrak{m}}$.

My question is: If the local blowup $Bl_{\mathfrak{m}_{\mathfrak{m}}}(Spec(R_{\mathfrak{m}_\mathfrak{m}}))$ is nonsingular, does that imply that the blowup $Bl_{\mathfrak{m}}(X)$ is also nonsingular? If not, is there an obvious counterexample? If so, can this be generalized to schemes with multiple isolated singularities or possibly to non-isolated singular loci?