I reluctantly say Yes.
From Wild Shape, it seems like your ability to speak is based on the capabilities of the beast form. You are correct to assume you would be able to speak. The comma, however, implies that you cannot cast spells regardless, as everything after it is essentially its own sentence.
You can’t cast spells, and your ability to speak or
take any action that requires hands is limited to the
capabilities o f your beast form.
From the look of it, the Awaken spell is a permanent transmutation. The "Duration: Instantaneous" means that the magic doesn't stay on the creature. Nothing implies in the awakened spell that anti-magic could dispel the transformation, nor could a dispel magic spell. This is important, because you can transform into a creature with Enlarge cast on it, but you do not get a version with the Enlarge spell actively cast upon it. Here, however, the spell was instantaneous and non-reversible. Assuming that we can agree that you could transform into a one-legged dog that you have seen, you would be able to transform into the Awakened Animal.
Casting Time: 8 hours
Components: V, S, M (an agate worth
at least 1,000 gp, which the spell consumes)
I will also note that Polymorph will remove your hand and speech options regardless of the chosen creature. So a polymorph monkey could not use their hands.
The creature is limited in the actions it can perform by
the nature o f its new form, and it can’t speak, cast spells,
or take any other action that requires hands or speech.
One issue that I cannot answer is what the new CR rating is for the creature. An increase in statistics would increase the CR rating, and I don't believe there is anything officially published about how Awakened affects the CR rating of a beast (Correct me if i'm wrong.) Because Polymorph and Wild Shape are CR dependent, this information is vital in determining the validity of this idea. You would have to speak with the DM to decide what the new CR rating is for the beast.