  The resonant scattering interaction between Ly$\alpha$ photons and neutral
hydrogen implies that a partially neutral IGM can significantly impact the
detectability of Ly$\alpha$ emission in galaxies. The redshift evolution of the
Ly$\alpha$ equivalent width distribution of galaxies thus offers a key probe of
the degree of ionization during the Epoch of Reionization (EoR). Previous
in-depth investigations at $z$ $\geq$ 7 were limited by ground-based instrument
capabilities. We present an extensive study of Ly$\alpha$ emission from
galaxies at 4 < $z$ < 8.5, observed from the CEERS and JADES surveys in the
JWST NIRSpec/PRISM configuration. The sample consists of 235 galaxies, among
which we identify 65 as Ly$\alpha$ emitters. We first measure Ly$\alpha$ escape
fractions from Balmer lines, and explore the correlations with the inferred
galaxies' physical properties, which are similar to those found at lower
redshift. We also investigate the possible connection between the escape of
Ly$\alpha$ photons and the inferred escape fractions of LyC photons obtained
from indirect indicators. We then analyze the redshift evolution of the
Ly$\alpha$ emitter fraction, finding lower average values at $z$ = 5 and 6
compared to ground-based observations. At $z$ = 7 we find a very large
difference in Ly$\alpha$ visibility between the EGS and GOODS-South fields,
possibly due to the presence of early reionized regions in the EGS. Such large
variance is also expected in the Cosmic Dawn II radiation-hydrodynamical
simulation. Our findings suggest a scenario in which the ending phase of the
EoR is characterized by $\sim$ 1 pMpc ionized bubbles around a high fraction of
moderately bright galaxies. Finally, we characterize such two ionized regions
found in the EGS at $z$ = 7.18 and $z$ = 7.49 by estimating the radius of the
ionized bubble that each of the spectroscopically-confirmed members could have
created.
