I had this problem when i was applying
although i did have Psychology, Sociology and Media studies, some universities require you to have one science (even though i was applying for psychology, they didnt include it in the science list)
I was looking at applying to Oxford for the PPP course and my rep said it would of been fine, i choose cambridge after anyways so im not sure what would of happened there, but my rep said it was okay!
Plus, if you look at Scottish universities they do MA/BA, the MA you do not need any science based subjects, yet with the BA you do.
St Andrews is a really good university for Psychology and you do not need any science subjects.
Its number 3 in the united kingdom for Psychology, after Oxford and UCL, so you might want to look into it? Basically im saying, its probably the best university you would get into not having science based subjects for Psychology
...however judging by your Oxford/UCL im guessing youd rather stay down south?
xxxx
{I should also add, i emailed st andrews before applying asking whether having an MA in psychology would ruin my changes of going into Clinical psychology later on, they replied saying that it would not as the course is the same basically)