This is Denis-Charles Cisinski's answer, given in the comments:
Yes, the map $\pi_0(W^{-1}C)\to W^{-1}\pi_0(C)$ is always an equivalence of categories (this follows immediately by comparing the corresponding universal properties). The same remains true (for formal reasons as well) if you look at the truncations of hom-spaces defined by $\pi_1$ and work up to an adequate notion of equivalence of 2-categories (in fact, you may as well truncate in dimension $n$ and get an equivalence of (n+1,1)-categories).
