Oh, bless her. My birthday's a few days later but even then, absolutely nobody has ever cared about it in the slightest. (My own mother once gave me tea towels she bought in the sales, with the price tag still attached.)
So, with that in mind, I'm going to disagree slightly with others - and, indeed, with what I'd normally say (which is that one year olds aren't remotely concerned with birthdays so why bother) - and suggest you might use whatever you do as a sort of audition process for finding something people will get on board with, so when she's big enough to care herself you have an established plan and it's part of people's mental calendar of stuff that happens.
The issue with trying to celebrate then is, firstly, as you say, everyone is probably hung over (well, the parents, anyway), but also they've all run out of money and probably made resolutions which preclude doing anything interesting. So, definitely nothing that requires them to spend much, and make sure there are options to avoid basically any imaginable foodstuff because the parents will all have suddenly decided they're vegan, or don't eat sugar, or aren't doing gluten, or whatever.
Assuming covid doesn't scupper things, I would suggest that the natural solution to all these problems is to find a lovely, child-friendly brunch venue - preferably one that's a bit woo in terms of menu choices - and just have a really nice, laid-back get together. Go for a Sunday morning vibe. That way you can make it something lovely that everyone looks forward to, rather than oh-good-grief-now-a-birthday-party-too?