I'm an A-Level Psychology/Sociology teacher, so I may be able to help
It is possible to qualify to teach A-Level Psychology, you'd need to apply for a PGCE in Further, Higher and Adult Education (I think there are about 12 institutions that do it, scattered around the country). This is what I did (in Sociology), however I wouldn't recommend doing it; I've had a nightmare, and have basically had to requalify because whilst the PGCE FE qualifies you to teach A-Level it doesn't (for some odd reason) qualify for you to do so in a school with a sixth form.
What I'd recommend instead is a PGCE 11-18 in Social Science (have a look at the GTTP website for uni's that do it, I know Manchester Met do, but am sure it's not a rare course). This will basically enable you to teach A-Level Psych (in fact, you'll get preference over people with a PGCE FE when you come to apply for jobs), but also to teach in secondary schools as it confers QTS. GCSE Psych is rare in schools, but you get around it by qualifying in another subject; usually Citizenship, PSHE and RE.
If your set against teaching schoolies, then I'd recommend applying for an A-Level teaching job and qualify "on the job" - although this is probably not the best route straight out of uni.
Feel free to PM me if that doesn't make sense.