Some might, others might not. Will be entirely dependent on the uni and course. Generally speaking the more "scientific" courses (e.g. at Oxbridge, UCL, KCL, Bath, etc) often require some formal qualifications in lab sciences and/or maths. Other ones may be happy with a more general social science background. Usually the ones preferring a scientific background are the much more research focused courses/unis mainly doing work in experimental (rather than clinical or applied areas of) psychology.
Bear in mind though in any psychology degree you will do a minimum amount of scientific and mathematical (mostly statistical) work, due to BPS accreditation requirements. So you should be prepared for that even if they accept you without any background in those areas already. I gather the stats work in particular can be quite extensive in most courses (the more neurobiological content will always have a minimum but how far beyond that they go really depends on the uni as I understand).