You're not rambling duck.
I doubt the doctor will see you without your DD but she can give consent for you to join her own appointment. No reason why you can't make an appointment time for her though and talk everything through together. She might need you as her mouthpiece in any event.
The work at Durham will not be a challenge for a student as clearly able as your DD is.
The key thing is for your DD not to struggle through this alone, however reluctant she is to talk. I think that's not uncommon, for young people with issues like this. In some cases it seems the more serious the issue, the more reluctant to talk. Really, the chances are that the university services at a uni like Durham will actually be in a far better position than your home GP to give meaningful help. Unfortunate though it is, they deal with cases like your DD's all the time and have done for generations - I can name a good handful of people at Durham that needed the same sort of support, just off the top of my head, and that was back in the days when people were far less open about difficulties. I'm not in any way meaning to trivialize the problems which are very clearly significant but I really do think Durham is one of those unis which can tend to attract students with these kind of issues, so their medical and pastoral support is very high quality, and readily given. For that reason alone she may be much better off staying - it gives her the opportunity to face down the problems rather than avoid them whilst at the same time getting top quality support. And if work slips while she gets treatment then I'm sure the uni will look kindly on repeating a year. To my mind that last would be a better option than returning home, and another void for a year - either way the overall time to get the degree takes four years, but the void way is more demoralizing. One friend of one of my DDs was in a very similar boat and put off uni for a year, then re-applied but didn't take the place, then dropped out the following year having started the course - she felt far older than the others on the course (but hadn't helped herself by opting for a Scottish uni). She's well in herself, but has a very limited life for someone so incredibly clever. That's what I mean about trying to hang on to the future.
Very best of luck.