Hi there! I'm a PBS student that has been through the first year in Cambridge.
It would be beneficial if you take A level statistics because most of the statistics we learn is a revision of that, with only a little bit of new material. However because the maths portion of the second PBS paper is not so much, you can get away with not having learnt that stuff before. However the pace of the lectures can be quite fast if you're not mathematically inclined, so it would be best to have done some maths before.
Chemistry and physics are definitely not needed. Biology though is very beneficial (again it has to do with the second PBS paper). It would help you immensely if you had biology, especially since you mention that you haven't taken psychology as an A level (I hadn't either!). But I understand that some students taking the course aren't scientifically inclined, so the only way you could get around it is to have basic GCSE knowledge at least.
They understand that people come from a lot of different backgrounds. There are some courses that are linked to the HSPS tripos if you don't want to do natural science papers. Social anthropology, biological anthropology, or archaeology are three very popular papers that people take and as far as I know, you don't need any prior experience with these subjects as they start totally from scratch.
For most colleges, you'll have to send in an essay. But because you don't do psychology, it would be best to apply to a college that requires interview only, like I did.
Computer science is an optional paper and I was the one unfortunate to choose it. Don't take it. Just don't. It's the hardest paper one has to do in this course and it all fell apart pretty quickly. :P