I do not think the LLM will meet your needs.
To qualify as a lawyer you need to have taken certain compulsory courses (the GDL) then take the legal practice course (LPC) then do a two year training contract. In those last two years, if you choose the right firm, you may be able to direct your career towards medical law.
Both the GDL and LPC are expensive and privately funded. The best thing to do is to get a training contract organised with a firm who will sponsor you.
Start by researching firms that do medical law (Legal 500, Chambers & Partners). Then identify when their training contract recruitment timetable is. You will then be able to organise your training contract applications.
Be warned though, that the training contract process is extremely competitive. You will need good A'level grades on top of your good degree classification.