If that's what has to happen for her to hold her mental health together then that is what has to happen. It sounds like she puts an incredible amount of pressure on herself if she has chosen to do 13 GCSEs and perhaps sees her worth as all tied up with how intelligent she is. It sounds like she may have perfectionist tendencies too?
Do you think in a couple of days you could suggest she just does English and maths (and possibly the GCSE's relevant to what she wants to do at uni or alternatively the ones she feels have gone well so far)? Definitely cut the number down by at least half though.
If her anxiety is this high with GCSE's then I would definitely not recommend A-levels as the way to go. A-levels are 100 times more stressful and the exams test such a small amount of what is learnt through the course, it is really intense. They really do not suit a lot of people, including intelligent people. I for one got CCE in my A-levels but later passed my MA(Res) with distinctions.
Once GCSE's are over, whatever happens, I would start putting the idea of an Access to HE course into her head. Keep mentioning that you've heard really positive things about them. Mention that you heard you could even get into Oxford with one. Mention that you've had a look and they are accepted in RG unis. Basically just drip feed the idea into her head - this is often a method I find that works with DS with ASD. He needs time to very slowly come round to an idea.
I also think she should think very carefully about unis. While the RG's might have a certain prestige, a fantastic researcher is not always a great teacher IME. Sometimes they're really not interested in teaching at all - just have to do it part of the job. When we went to visit uni's the one I was most impressed by the lecturers and clearly had the most interest in the students, their progression and their welfare was not Southampton or Exeter (RG's), it was Surrey.
At Surrey the department head mentioned the projects of individual students she had been working with. At Southampton the lecturer opened with 'don't expect me to know any of your names next year because there'll be 300+ of you'.
I would also say that as an autistic student with MH struggles I would really recommend she consider a uni close to home that gives her a chance to come home at weekends at least, if not live at home and commute in. Also look into their reputation for pastoral care and what they will put in place for students with ASD. There are things that are much more important than a bit of prestige in her situation IMO.