We know that if $V=L$ holds, then $|\cal{P}(\omega)|=|\cal{P}(\omega)\cap \textrm{L}|=\aleph_1$ whereas, in the presence of a measurable cardinal (in fact, even Ramsey) $|\cal{P}(\omega)\cap \textrm{L}|=\aleph_0$. I remark that the cardinalities are of course computed in (the corresponding) $V$.

The first is just the fact that the constructible universe satisfies CH, while the second has to do with the fact that in the presence of a measurable, $\omega_1^{L}<\omega_1$ i.e. the existence of large cardinals makes the relative $\omega_1^{L}$ "drop" below its "maximum possible" value (which is attained, if you want, in the "extreme case" when $V=L$).

My question is, what can we say, in general, about the beaviour of $\omega_1^{L}$ given axioms of increasing strength above (or equal to, in strength) $V\neq L$? In particular, what happens if we just assume $V\neq L$?