Grades for universities

For some of the universities I have been looking at they ask for grade 6s in Maths and Science, however I have grade 5s. Is there still a chance that I could be accepted?

For what subject(s) and how competitive are the unis? It really depends unfortunately

I've been looking at the top 20 unis, especially for Psychology. (Cambridge, Oxford, Warwick, London, St Andrews, etc.)

GCSE grades are non-negotiable - they are about the basic skills required for that course, typically Maths.

Applying without the right GCSE grades is one of the quickest ways to get an immediate rejection.

what do you think about my theory that with a good a level maths grade they wouldn't care? Admittedly oxbridge would probably but I think most unis would be reasonable and recognise that the maths skills are clearly there?

For some of the universities I have been looking at they ask for grade 6s in Maths and Science, however I have grade 5s. Is there still a chance that I could be accepted?

OK, if you're taking maths a level and do well (A/A*) on that then you should be in with a shot as that shows you have the maths ability to keep up with the course. Oxford and Cambridge get a lot of good candidates and basically run out of ways to differentiate between them so tend to put more emphasis on GCSE results than the others (there are only so many A*A*A*A* top grade STEP 10/10 interview maths applicants they can take so imo they just start making up stuff to care about! Was my 9 in spanish really helpful to a physics degree).

Do any of those unis ask for entrance exams, interviews, submitted work? If you're missing part of the offer (quite an insubstantial part though) the best bet is to outperform the rest of it so they have no doubts about your academic ability, so you want as many opportunities as possible to show what you can do. The more components of the application there are the less each one matters.
My friend applied for an art degree which asked for 6 in gcse english and a 6 in another humanity gcse, plus AAB at a level with an interview round
She got an unconditional after applying with 9 in gcse english, no other humanity gcse, A*A at a level and B at AS and a predicted D in art foundation (and probably smashing the interview). You see how despite missing parts of the standard offer she outperformed heavily in others (GCSE English and having an extra qualification with a high predicted).

Art degrees are more flexible than psych, but you might still be in with a shot. You only narrowly missed the GCSE grade. Email a couple of unis (maybe a long shot like oxbridge, somewhere in london, st andrews, and a lower russel group) and see what they think. Sell yourself in the email, don't be afraid to put some spiel in about how you really do think you're capable of the maths on the course. Also consider addressing your GCSE grade in your personal statement. I think psych degrees are fairly stats heavy? If you don't do A-Level maths (or even if you do) could you take a MOOC in stats to show the unis that you're committed to keeping up with the maths requirement?

Also if I remember right a 5 is the highest grade you can get on foundation maths? you didn't happen to do that did you, because that might work in your favour if you can say you believe you would have got a 6 if you'd been let take higher...