I don't understand why you wouldn't be 18, did you skip a year or something ? So you're just turned 16 ??
Perhaps either email/ring the universities, i don't think it would be too much of a problem, and if you cant, taking a gap year will be good for you, you'll mature etc
Regarding, Cambridge don't put your hopes on it, its very hard to get a place, but you seem very well prepared (you mentioned you've done extra curricular etc) but alot of it is still down to luck (and bmat arrggh)
Its a bit harsh for you mum to say she wont support you, but, im sure she'll come round eventually, she may just be saying it, thinking it will motivate you further. No medical school is a bad one, its not like law where getting a job is based on where you studied, just read up on the course structure (based on what/way you wrote, you seem more suited for the traditional course, ie more academic) and look up on the location.
At the end of the day, its your choice, not your mums, apply for where YOU want to, not where she says !