It probably helps that Belle doesn't do a whole lot of singing in the original. She sings in only a couple of songs ("Belle" and "Something There") and only about half of each. In total, it's less than 2 minutes of singing. So, it's not like Watson has the burden of carrying the soundtrack like a trained singer. Collectively, the other singing parts are a lot more important, so, as long as they get professional singers for those, they can probably get away with an amateur like Watson, especially when she and a vocal coach can practice and perfect what is a relatively small number of singing lines. It probably also helps that they're songs that are well known to her and burned into her brain.
EDIT: I'll add that I'm not sure if Belle sings more in the musical and if they'll incorporate any of those songs, or if they'll write any new songs for this new version. It's possible that she'll sing more in this than the original film. Even so, I think that the point remains that it's still largely an ensemble signing cast, not one that needs to be carried by Watson.