There are consistency results due to Shelah and Gorelic for the negative answer. Shelah has proved the consistency of $GCH+$there exists a Lindelof space with points $G_\delta$ of size $\aleph_2.$ See Shelah's paper ''On some problems in general topology''. Gorelic has extended this result to get $CH+2^{\aleph_1}$ arbitrary large+ there is such a space of size $2^{\aleph_1}$.
It is also consistent with $GCH$ that there exists a Lindelof space with points $G_\delta$ of size $\aleph_\omega.$ See ''A topological application of flat morasses'', Fundamenta Mathematicae, 194 (2007) pp. 45-66.
On the other hand Shelah has proved the consistency of $CH+$ there is no Lindelof space with points $G_\delta$ of size $\aleph_2.$ See Shelah's paper ''On some problems in general topology''.