I rule "yes, the skeleton can speak" on this issue. The spell does say "you can cast this spell on a corpse that has been deceased for any amount of time". If corpses reduced to a skeleton could by their nature not speak, one would think that would present a hard limitation on the spell. A distinction between "having a jaw and teeth" versus "having a mouth" doesn't appear in the spell.

There's lots of examples of skelatal-appearing dead speaking in fantasy and gaming. In D&D, somehow spectres, ghosts, liches, and whatnot can speak even though they don't have physical muscles to do so. This is a Necromantic spell, after all.