I have a 2:2 BSc honours degree. UK. Could I do a Phd?

You don't need a Master's, you can go straight from Bachelor's to Doctorate. In theory, at least. The fact that you have a 2:2 (which is not just a 2nd class, but lower 2nd) isn't going to impress many universities too much. Even if they don't actually require a 2:1 or better (which most do), they will ask the obvious question, ie. if you couldn't do better than that at undergraduate level, how are you ever going to cope at postgrad research level where things get more difficult by orders of magnitude. And whether your answer is effectively that you're not very intelligent or you're not a hard worker, I'm not sure either will sound very convincing. Obviously any respectable university will want to choose the best possible candidates for their PhD programme, so why should they choose you over someone who got a 1st?

In science subjects, usually they aren't interested in a Master's first. The problem is, and this was the killer for me with a 2:2 in physics, is paying for it - it's much harder to get funded if you don't have a first or a 2:1.