Subjectively... Yes.
Reincarnate states:
For nonhumanoid creatures, a similar table of creatures of the same type should be created.
A reincarnated creature recalls the majority of its former life and form.
Awaken states:
Its type becomes magical beast (augmented animal).
What does all of this mean when we put it all together?
The creature, would be reincarnated as another magical beast (augmented animal). It would remember that it could speak in its former life. Since it would still be a magical beast (augmented animal) it would retain its ability to speak in its new form (depending on the new form - griffons can't speak, for example):
The reincarnated creature gains all abilities associated with its new form, including forms of movement and speeds, natural armor, natural attacks, extraordinary abilities, and the like,
It would not, though, gain the new language of its new form automatically:
but it doesn’t automatically speak the language of the new form.
Many magical beasts speak.
Blink Dogs speak, and even have their own language.
Worgs speak, and even have their own language. Some magical beasts, however, such as a Griffon, specifically states that it can understand but can't speak.
If reincarnated into a magical beast that can speak, there is no reason you can't speak the languages that was formerly known.
Food for Thought
Since Reincarnate states:
For nonhumanoid creatures, a similar table of creatures of the same type should be created.
A table of creatures to be reincarnated into should all be magical beast (augmented animal), not animal, not humanoid, not even nonsub-typed magical beasts. That would mean, that it would reincarnate into similarly "awakened" animals.