Actually they do... I'm surprised so many people have said this isn't true without checking it. (I applied - and got in - last year so I remember this being a requirement.)
http://www.psy.ox.ac.uk/undergrad/ugentryrequirmentsAlso:
http://www.psy.ox.ac.uk/undergrad/selection_criteriaI'm not saying they'd never make an exception, but they certainly list it in their requirements.
OP, I'm sorry to say this, but with your A level subjects and GCSE grades, it doesn't sound like you'd be likely to get an interview at Oxford. (See the first link for "an idea of what subjects and grades we would normally expect from candidates applying to do a Experimental Psychology or Psychology & Philosophy Degree at Oxford".) When I applied, I thought pretty much everyone would get an interview for EP, but when I went down, one of the helpers said that only 12 out of the 30 applicants to that college were invited to interview for the 4 spots available. I don't know how typical that is though. I know there were some people last year on here with very impressive subject related extra curriculars/experience and very good (but not the absolute best) subjects and grades who I was surprised didn't get interviews.
I know you said you're definitely applying and that's fine if you can't bear the thought of not at least giving it a shot. Maybe it would be an idea to email the admissions department though and ask whether they'd consider you. That way if they say no you've at least given it some kind of shot but you haven't wasted a spot on your application that could prove very important. I don't say all this to be harsh. I just think it's best to have as much info as possible.
Have a good look at the requirements for all the places you're applying to and make sure you meet them. It's generally not a good idea to apply to places where you don't meet the requirements as it just makes the admissions tutor's job very easy when he's sifting through the thousands of applications and wondering which to reject. Applying to one long shot, two at about the right level and two with lower requirements is a good idea, or you could apply to 5 with progressively lower requirements which is sort of what I did, although not really intentionally.
Some places you could look at that have particularly good psych departments but are either less competitive or have lower grade requirements are York (lower grade reqs but very competitive), Nottingham (less competitive, also has option of psych with cognitive neuroscience which has a lower offer), Birmingham (I didn't apply there but I think it's somewhere inbetween for competitiveness), Cardiff, St Andrews (high grade reqs but in the middle of nowhere so not as competitive as it could be), Glasgow, Kent, Bangor (very good psych dep and like 1/3 of their students do psych but very low entrance reqs - again in the middle of nowhere but a nice area) and Sussex.
These places have particularly good reps for psychology but aren't necessarily as well known overall. (Also Bath and Bristol but they're very competitive *and* have high grade reqs. Oh and to some extent Exeter. That's worth looking at.) Durham and Warwick are more the other way around. Obviously this is pretty subjective though and not an exhaustive list. If you're not planning on a career in psych, then the overall rep might be more important to you.
There's no point in applying to places you wouldn't go to even if you had no other offers, but having a spread of unis with different requirements is a good idea. You've picked four of the most prestigious unis in the country (not for psych, but in general) and then one considerably lower. If you get rejected from the first four, you might find yourself wishing you had something inbetween.
I hope this helps. Good luck.
Edit - Also, you might want to look at how much emphasis the websites put on sciences. A lot want a minimum of one science subject at A level and I seem to remember somewhere requesting two. They generally accept psychology for this but when it comes to making offers, they often seem to favour those with more sciences. I'm not saying avoid these places, just be aware that they could be more risky. Bath's course appears to be more social-sciencey (as opposed to experimental/natural science) than most (and is why I didn't apply there). I don't know if that makes it more or less appealing to you? Oxford, York, Bristol and UCL particularly emphasise the science aspect. Not so sure about other places.