I graduated with a similar degree a couple of years ago and took out a graduate loan to pay for an introductory film making course in London. Cost about 800 quid at the time and if i'd lived in London it would have been ideal (although it did give me some practical experience for my cv and showed i'd got involved) I didn't live in London (still don't) so i signed up with shootingpeople and was contacted to help out as runner on a short over one weekend.
I should add that this was over a couple of years and also involved lots of dull temping and talking about working in film & television instead of doing it. If you're in London (Manchester, Birmingham, Bristol etc) clearly there are more opportunities – If you like London as a place to live – it's personal preference.
I would say just jump in and get involved. Every project you work on you gain more experience meet more people and gather momentum.
Ultimately my wife worked with someone whos ex worked as a producer and i got their email address and got in touch. Attached my CV had an interview, got a few weeks work experience, then a few more and now i'm full-time. I work as a production coordinator by day and try to work on my own project after work and at the weekend to satisfy my creative aspirations.
Just show them that you are enthusiastic and capable and willing to start early and stay late.