I am running Lost Mine of Phandelver and my party just destroyed the Flameskull for the 1st time.
Since it will rejuvenate in 1 hour, my question is: does it recover all spell slots used or not?
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Rejuvenation. If the flameskull is destroyed, it regains all its hit points in 1 hour unless holy water is sprinkled on its remains or a dispel magic or remove curse spell is cast on them. (MM 134, emphasis mine)
This trait says nothing about spellcasting, it only states that hit points are restored. So that is the only effect it has. It is still the same skull, so the spells already used up return only next day.
I would say that it does not recover its spells because that usually is taken up by a Long Rest. I'm not familiar with the module itself, but if it had Warlock traits to it, perhaps then you could recover some spells since the 1 hour restoration would fit into the Short Rest category.