Hey there I applied for Cambridge, Bath, ucl, lse and Exeter this year for psych and I got offers for all of them! I applied with maths, further maths, Spanish and Psychology a level (altho now I take further maths AS) so not doing biology is definitely not going to completely reduce your chances of getting into uni! Whilst for some unis it does say that maths and bio are preffered, it is not the end of the world if you don't do them, and besides, you already do maths so that aspect is already covered. As long as you apply with strong predicted grades and a personal statement that shows your genuine interest for psychology, you probably have as much of a chance as everyone else! The only thing to be careful about is that as far as I'm aware there may be a few Cambridge colleges that definitely want people to take bio, but those aren't in the majority. Overall, you definitely should not stress about not doing bio! (just keep in mind that there are lots of other aspects that come into uni decisions as well as subjects)