How Does FM Get Around the Gabor Limit?
This suggests it's impossible to perfectly determine the instantaneous signal even in a theoretical channel with zero noise


I'd turn this around and say it's impossible to instantaneously determine the signal. 


  one has to assume that the signal is changing slowly (band-limited). How is this overcome in practice?


In practice, our message signals are bandlimited, so this is not a difficulty. In fact, our message signals generally have much less bandwidth than the carrier.

To approach your theoretical question, is a band limit strictly required,  Imagine trying to modulate a 1 Hz carrier with a 1 MHz signal --- the result would be unusable. So in fact there must be some kind of limit.