We study Si-SiC core-shell nanowires by means of electronic structure first-principles calculations. We show that the strain induced by the growth of a lattice-mismatched SiC shell can drive a semiconductor-metal transition, which in the case of ultrathin Si cores is already observed for shells of more than one monolayer. Core-shell nanowires with thicker cores, however, remain semiconducting even when four SiC monolayers are grown, paving the way to versatile, biocompatible nanowire-based sensors.