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The countability assumption cannot matter. The reason is that any uncountable model $M$ is countable in a forcing extension of the set-theoretic universe. If $M$ has a proper elementary end-extension in the original universe, then this end-extension still exists in the forcing extension. So we may apply the theorem as you have stated it in the forcing extension in order to deduce that $M\models\text{PA}$. But the question whether $M$ is a model of PA or not is absolute between the set-theoretic universe $V$ and its forcing extensions $V[G]$, and so $M\models\text{PA}$ in $V$, as desired.