Each of the following implies that (the true) $\omega_1$ is inaccessible in $L$, and hence that there are only countably many constructible reals:
- The proper forcing axiom
- There is a Ramsey cardinal
- $0^\#$ exists
- All projective sets are Lebesgue measurable
- All $\Sigma^1_3$-sets are Lebesgue measurable
The mere existence of a nonconstructible set, or even a nonconstructible real, does not imply that $\omega_1^L$ is countable. There are many forcing notions in $L$ which do not collapse $\omega_1$: adding one or many Cohen reals, destroying Souslin trees, etc. Each such forcing (over L) results in a model where $\omega_1=\omega_1^L$.
In fact, "Martin's axiom plus continuum is arbitrarily large" is consistent with $\omega_1^L=\omega_1$. (And also with $\omega_1^L<\omega_1$.)