This is not an answer about the unitary case, but for an analytic family of selfadjoint matrices, the eigenvalues are analytic, according to Rellich's theorem, as described in the Linear Algebra book of P. Lax.

This is not an answer about the unitary case, but for an analytic family of selfadjoint matrices, the eigenvalues are analytic, according to Rellich's theorem, as described in the Linear Algebra book of P. Lax. Edit: but even if such a result holds for the unitary case, an analytic eigenvalue $\lambda(t)$ might not be expressible as $e^{i\theta(t)}$ because there are maps of $U(n)$ into $S^1$ which are not nullhomotopic.