Hmm, not promising beginnings. I'm sorry you're going through this.
Most people are a little apprehensive about beginning university, but having mental health conditions and suddenly being made self-sufficient would really throw you in the deep end. It happens to a lot of people when they start university, and it can take people to a pretty bad place.
I would seriously consider whether you would be better off deferring until next year, sorting yourself out and taking a gap year.
I'm sorry you've had such a negative experience on the NHS. It is very much a question of pot luck and a number of people are unfortunately put off by the abominable state of mental health care in this country, and never seek help again. Don't be one of those people. If you want to get better, which I really feel like you do, then you have to be determined and persistent. You've had a bad counselling experience - so seek out a new counsellor. And keep on seeing your GP communicating how desperate you are feeling. Eventually they'll have to refer you to a psychiatrist if all other plans of action fail. You can get good help on the NHS for mental health but you have to persist like crazy.
Please, don't just accept you are ill. You may have had a rough time recently, and are feeling pretty friendless, but that happens to the best of us. People regenerate themselves, build new lives. Why can't that be you?
Hope you feel better soon.