The idea of using Polymorph to get around the limitations on Awaken has been covered before. It has a bunch of issues, having to do with the fact that eventually you’ll want to unpolymorph the thing back into its original state and (preferably) keep the int boost and the free language, and there’s a lot of reasons that might not work the way you want. Find Familiar, though, gives you a critter that basically is a low-CR beast, other than the fact that it’s not actually a beast, so there’s much less incentive to change it back when you’re done. You can’t normally abuse True Polymorph to permanently upgrade your familiar in a meaningful way, because you can’t increase CR with that spell… but changing it into its actually-a-beast version, or into some other beast of the same CR should still be doable, and that seems like it would make the creature a valid target for Awaken. At that point, it seems like you could just… leave it as it is. It would be a beast instead of a celestial/fey/fiend, but that seems like a small enough price to pay for cranking the int up to 10 and giving it the ability to speak a language. There are a number of other useful features of familiars, but those all seem to be facts about what the caster can do, rather than things that would be disrupted by True Polymorph.
Basically, this looks like cheese. It’s a way to permanently upgrade your familiar with spells, and it seems pretty clear that the authors have made efforts to try to prevent exactly that. At the same time, as far as I can see, it’s viable-within-the-rules cheese. Is there anything I’m missing here?
The argument "it’s obviously cheesy so as DM I wouldn’t allow it" is not a acceptable answer to this question unless you can also find some in-the-rules way to challenge its validity. If doing this would cause you to lose out on any of the standard features of a familiar, that would be useful additional information.