Humanities, Philosophy and LawWatch
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You can become a barrister or solicitor in England with any first degree. To become a barrister you would need to do a Graduate Diploma in Law (GDL) conversion course, to become a solicitor you now need only sit the Solicitor's Qualifying Examination (SQE) and don't need to do a GDL.
There aren't any similar programmes to the JD/MD in the UK; Cambridge I believe used to potentially allow students to do their intercalated course over two years in Law to get a qualifying law degree, but from what I've heard that's no longer permitted
Because of how the medical and legal profession post-graduate training is arranged here they're not really compatible as such. To qualify as a lawyer in the UK you need to undertake a period of supervised training in an approved post, and for the barrister side complete a vocational course (the BPTC), before you are qualified to practice yourself. Likewise for medicine, medical graduates aren't fully registered and able to legally practice independently until after completing the first year of the foundation programme (and they need to complete the second year to then apply to specialty or GP training anyway). Although in medicine they can do their training less than full time, I'm not sure any such possibility exists for law (on either the solicitor side or barrister side). In the US, on the other hand, my understanding is MD grads are immediately full registered and qualified doctors (although obviously haven't done specialty training), and JD grads can practice in their given state once they pass the bar exam without any specific training requirements otherwise.
While it's certainly possible to go into the legal profession after medical training (it's actually possible to become a lawyer after doing any degree in the UK), if that's the goal at the outset, it seems a bit besides the point to go up against the enormous barriers to entry for a medical degree, then do a 5-6 year degree, before even being able to consider doing their legal training (and for the barrister side they'd need to take another year to do a graduate diploma in law)...compared to just doing a 3 year law degree (or any standard 3 year BA/BSc for that matter), which is considerably easier to get in (at least outside of LSE and Oxbridge), and much more straightforward.