I'm in the middle of my LPC at the moment. My undergrad is in a biomedical science (think genetics/biochem sort of area) then I worked for a couple of years in software before deciding to start my GDL.
It's tough. Very tough. The market for new lawyers is not a pleasant one and a considerable number of people from my GDL realised that and decided not to do the LPC until they'd secured a contract.
Ideally you'd get a TC before starting but more and more people are now self funding the GDL because they just want to get started on the path and TCs are so hard to come by. A lot of it depends on where you are and the firms you're willing to apply to. If you're in London there's no shortage, but I'm in the NE and choice is exceptionally limited if you want to work in a well recognised firm that will lead to decent enough pay. There's a handful of firms and they only have up to maybe 10 spots a year each. Some have less than that. There are plenty of smaller high street firms too but a lot of them only have one or two training spots and hire straight from the LPC so you'd need to self fund in that instance.
Although experience if you're converting isn't the be all and end all, I'd advise getting some anyway for your own piece of mind. I've been a paralegal for two and a bit years in a big firm, and I'm doing it alongside the full time course. It's helped a lot because at least I know what I'm getting myself into and what area of law I actually want to qualify into. There's a lot of people who have some basic court experience or a week here and there but in reality they have no idea how what we're learning will apply in the workplace, how hard the slog can be, and sometimes how monotonous it can be at the start. It's a scary concept! Some of them have become pretty disillusioned when they've come back from longer work experience or vacation schemes.
Firms run vacation schemes for TC candidates which are a great way to gain experience, insight into the firm, and sell yourself to the recruiters, but it isn't easy to get places on them as the selection process is the same as for TCs. If you know anyone in law that could help you get some additional experience it's definitely worth it.
It isn't all doom and gloom though! I genuinely find my job fascinating (I do commercial work) and enjoy it. Yes it can be a bit dull but there's always something interesting around the corner. You have to be willing to throw yourself into it if you want to progress though. The partner that heads up my team made partner in about 10 years which is insanely fast, but he lives and breathes work and rarely leaves the office before 7pm, then will still occasionally be sending emails at 11pm!
I agree that converting from a medical setting works to your advantage. You've been in a workplace, before which many of the fresh-out-of-law-degree candidates haven't, and you have genuine experience in the area which is worth it's weight in gold to be honest. More and more trainees seem to be people that have converted either from a non-law subject degree, or from a previous career, because firms recognise that the diversity is beneficial.
You should look into the CiLex route too - you can't become a solicitor without a law degree/GDL and LPC, but you could become a Chartered Legal Executive. They are still lawyers, the pay grade depends on the firm, but training and education is cheaper overall I think and it's not as competitive. It would also be something that's easier to fit around part time work. Rather than a TC you can apply to be a paralegal and gain experience to boost your portfolio that way instead. Its certainly worth taking a look at.
Whatever you decide, good luck!!