I graduated last year with a 2.1 in Accounting & Finance from a very average University. Aside from the degree, I also have average A-Levels and GCSE's. I can safely say that the quality of University makes little to no difference when applying for ACA training contracts, as employers are more keen for quality A-Levels and GCSE's (more so A-Levels).
ACA exam results are closely linked to A-Levels. Therefore, in theory, if you did well in A-Levels, you are likely to do well in the ACA exams.
I work at a smallish firm with 9 partners and 3 offices. It's all pretty hands on and i'm already compiling sole trader and partnership accounts, even though I started a month ago (start chartered in Sept). One of my close friends started work at PwC last year and its completely different, less practical and more of a name than a face (less friendly etc). I was laughed at by several graduate finance recruiters when I said I was looking for an ACA contract with BCC at A-Level. I managed to get the contract by researching the firm extensively and showing this during the interview, it was clear to see that my results didn't accurately demonstrate my capability. Also, having a quality CV helps!
Obviously the money is going to be higher at one of the top 4 (during training) and it's always going to be a bonus having one of those firms on your CV, but in my opinion you're more likely to struggle in an accounts position once you have qualified.
EDIT: Also, even though the big 4 are pretty stringent with UCAS results, they occasionally search for graduates with low points, so keep checking recruitment pages and job sites. I managed to find an audit position in pensions at Ernst & Young, required a 2.2+ at University and stated no specific requirement of A-Levels. My results are distinctly average yet they offered me an interview! In a nut shell, never give up.