Well, firstly if you want to become a psychiatrist you'd need a medicine degree, which takes five years. You'd need four solid science subjects to do that, with a ton of work experience, fantastic GCSE's, and to get at least AAA or so. Wouldn't recommend it, personally.
Otherwise, if you want to go with clinical psychology (which is what it sounds more like) then you'll need to take a psychology degree first, as I'm sure you've worked out. It's not hugely important which university you go to for this, so long as it's BP accredited. You'll want to take one science-y related subject at least, whether that be psychology or biology, or a 'solid' subject that maybe involves essay writing, such as Philosophy or English Literature. A lot of your degree is based on statistics, so having a good (C+) GCSE Maths grade is essential, but don't worry about it too much - I'm useless with that sort of thing and managed to get a 1st with some hard work.
Hope that helps.