Whether $\mathcal P(\omega_1)$ is separable is independent of ZFC.

If $2^{\aleph_0} \neq 2^{\aleph_1}$ (which is consistent with ZFC -- it is implied by CH for example), then $\mathcal P(\omega_1)$ (which has size $2^{\aleph_1}$) is larger than any countably generated $\sigma$-algebra (which has size at most $2^{\aleph_0}$).

On the other hand, $MA + \neg CH$ implies that every size-$(<\!\mathfrak{c})$ subset $X$ of the real line is a "$Q$-set", which means that every subset of $X$ is a relative $G_\delta$ (a countable intersection of open subsets of $X$).
Suppose $X$ is such a set with $|X| = \aleph_1$. There is a countable collection of subsets of $X$ (namely, the basic open sets) that generates the $\sigma$-algebra $\mathcal P(X)$. Re-indexing the points of $X$ with $\omega_1$, we see that $\mathcal P(\omega_1)$ is countably generated.

The same argument shows that $\mathcal P(\omega_2)$, $\mathcal P(\omega_{42})$, $\mathcal P(\omega_{\omega^2+137})$, and every such set in between can, consistently, be countably generated as well. All you need to do is live in a model of set theory where Martin's Axiom holds and the continuum is at least $\aleph_{\omega^2+138}$.