Really depends on what you want to do afterwards.
For certain paths Oxbridge gives you an advantage or opportunities (e.g. Law, Media). It's also good for impressing strangers at social functions.
However, in the world of psychology it's far less of an factor. For a research career, it's more about academic kudos, which means scholarships and awards are more helpful and accessing research networks (which are good at both Oxbridge and St Andrews). It's not something most DClinPsy selectors care about if you wanted to do clinical psychology, and instead it's your clinical experience and your overall ability that they often test for at interview, but scholarships and publications do help you stand out from the pack. For other psychology routes its more about experience, supervision and developing skills.
The other things I would bear in mind is the advantages of being a big fish in a little pond. The downside of being at Cambridge is that you are far more likely to get lost or overlooked (because your are up against fairly formidable competition) than at a smaller or less focussed institution. People have written about this, but being the 'star' of a smaller institution where you can stand out win awards, make links with staff/mentors and develop your own opportunities, rather than being 'generic grad #32345' from the University of Impressing Your Mum's Friends. From my experience in both types of environments, I reckon it is probably one of the helpful thing to consider.