I don't sell the polices so I can't really help you with that because there are so many varieties of polices and add-ons and all sorts. But to be honest, since you already have a history, privates insurance companies won't let you claim for this because it's a pre-existing condition (unless, again, you work for a big company with a bespoke plan). You CAN however just pay to see someone privately as you already know.
What Uni will you be at? Most Unis have good health clinics. If you bring your medical records and/or a letter from your currect psych, you can then get a referral to one in your area. Many specialists will treat privately AND under the NHS so there's no reason that I am aware of to automatically rule out NHS care for this. So you might want to see someone privately first, then check out the NHS care and if that works for you, then you wont' have to pay for it! Otherwise, you can just budget in the fees for private visits.
Also, you might want to do some research into your prescriptions. As far as I'm aware, there are differences in what is prescribed in the US vs UK. I know, for example, that birth control can be an issue for some people.
I'm sure that there will be loads of private and NHS psychs in Bristol and WSM. Try
www.yell.com and see what you get that way.
Good luck!