Of course you can! Ok, you might struggle to work for a top 5 firm, but you certainly can. As a mature student you probably bring lots of life experience with you - so I would play on that if I were you.
Aim for regional or high street firms…
Get as much experience as you can and apply like mad. You can start applying now, though I would get a move on as you only have a few weeks left. You usually apply two years before you need to start.
Unfortunately, they dont always tell you that at some uni’s!