As far as I'm aware, she can't just "break you up" and forbid contact. You and your gf are both at an age where you can legally consent so you didn't do anything wrong. Sure it's understandable a parent will be protective of their daughter with guys but, at the same time, if she thought that you'd date and not end up doing anything sexual then she's an idiot.

I'd say keep in touch with your gf on the down low after things cool off (maybe hang out with mutual friends around) until her mum has grown the fuck up. As for your parents, I'd say that's down to you to decide. Honestly I'd maintain that since you didn't do anything illegal or anything then you are not obliged to tell your parents. In my mind sex is a private matter and as long as it's above board and kept to yourself then it's totally your business.

That being said, if your parents are not likely to similarly overreact then it'd be worth telling them from the standpoint of complying with your gf's mum and respecting her wishes. If they're in contact or something your parents may even be able to discuss the issue with her and talk sense to her.

Parents are overprotective by nature but ofttimes (like this) it's totally unnecessary and they just can't handle that their little darlings are experimenting with sex (as horny, hormonal teenagers tend to do) and forget that they themselves probably did at that age too.