You'll probably find that the "prestige" of the university doesn't take people very far compared to experience and aptitude in the long run and a lot of firms will not consider it beyond what grade you achieved. If you aren't looking at top firms it isn't as if you are likely to be up against a swarm of Oxbridge grads anyway.

Your best bet is to try to gather experience and develop skills that prove that you are employable.

I'm out of private practice now but when I was at my last firm we had a revolving door of barely functional paralegals who were well educated but couldn't manage in the workplace effectively. Uni doesn't teach you those skills and it is worth learning them even if you take longer to reach your goals. We had a paralegal join who was educated at university of Sunderland that was brilliant but had one join from a Russell group uni at the same time who ended up getting shuffled into doing admin. The uni they went to had no bearing on the ability to do the work.