Having just completed my graduate entry UCAS cycles I hope I can offer you a bit of hope. The first thing to do will be to find the universities that don't mind which subject you have done at University, as those that do will no doubt want some kind of life sciences. The second thing will be to find those that have no A level requirements that you don't meet; I know that all seems patronisingly obvious but if you choose the right uni's to apply to you will have as good a chance as anyone.
From my memory (of where I applied), Warwick would be a really good bet they only want a 2:1 (and couldn't care less if you have a 1st/2:1) from any degree and don't look at A levels or GCSE's so that would be a great option for you.
At graduate entry the UKCAT (or GAMSAT depending on where you apply) is often a huge factor for interview (although Warwick have changed the policy somewhat), although this is depressing if you don't do well, if you get a high score (high 600's, 700 +) you suddenly have a really strong application.
Obviously work experience at graduate level is key and ideally the will want to seen some kind of care in the community, voluntary/paid whatever, although people certainly get in with all kinds of backgrounds. Also like you say some kind of shadowing would be ideal, overall you simply need to have some experience and in your PS demonstrate exactly what you learnt from it and why it makes you suit bale for medicine (2 pieces of w/e well discussed in terms of skills learnt and how it prepares you is infinitely better than a vague list of 10+ pieces of w/e).
Sorry for the essay but in summary:
1). Apply to the Uni's that don't care about a degree not that ones you just like even though you don't meet the entry requirements.
2). Try and get a really good UKCAT score as it will definitely help.
3). Get some good work experience and discuss it in a meaningful way in your PS
4). Send it off, fingers crossed and good luck
Hope this helps