I'm in a similar position (2.1 science degree, not embarrassing but not amazing A levels), although I graduated three years ago and have been working for a science magazine since.
Looking at London-only unis you've got Bart's, St George's, and King's - Imperial stopped their four year course and UCL don't offer it. So you've got at least one non-London option.
St George's require the GAMSAT, which is a horrific five hour university-level exam in biology, chemistry, physics, and critical analysis. UKCAT is much more straightforward, but only relevant when applying to King's and Bart's.
Without stellar A level results, your application will really live or die by your entrance exam results and work experience/personal statement. I suggest you spend some time looking at the UKCAT, BMAT, and GAMSAT, and think about which would suit your strengths.
GAMSAT will require months of dedicated study, and unless you do it in Ireland in March, you won't know your results until after you apply in 2016. However, GAMSAT unis tend to have fewer applicants (they are intimidated by the exam!), so are a bit less competitive. BMAT is a bit less scary, but you only sit the exam after the UCAS deadline so you could have a bad test day and ruin your chances. UKCAT is easier although it requires a lot of concentrated effort in the weeks leading up to the test, but at least you get the results on the day.
Once you've picked your entrance exam strategy, you need to make your work experience watertight. You'll need sustained (months or years) paid or voluntary experience in a care setting (eg, befriender, hospice volunteer, healthcare assistant, first aider). You'll also need work experience in a hospital or GP practice, at least a week although two weeks is ideal.
While you're still at uni, it's worth cultivating a tutor or lecturer so you know they'll remember you when it comes to refereeing time!
But if you spend the next year wisely, you'll be in a great position next October.