To become a biomedical scientist in the NHS, you need two things: HCPC registration, and an IBMS accredited degree. If your degree is accredited by the IBMS but does not provide HCPC registration, you need to look for MLA roles that will allow you to develop the portfolio needed for HCPC registration. If your degree isn't accredited by the IBMS either, you need to have it assessed by the IBMS, and they will either determine it can be accredited retroactively (pretty rare), that you can achieved accreditation if you undertake further top-up modules with them (which can be expensive and time consuming), or that there isn't enough common content with the IBMS accreditation requirements and you will need to complete and entire new degree.
I would suggest in the first instance you find out if your course is IBMS accredited or not if you want to go the BMS route. For the clinical scientist route, I'm not as familiar with the recruitment process; I think you apply to a grad scheme and then progress internally through that. You'll need to do a bit of research into that option if you want to pursue it.
For medicine (graduate entry or otherwise) you will need to undertake relevant work experience and reflect on it (although for next years application cycle I believe most medical schools have suspended work experience requirements for standard entry medicine at least, due to the coronavirus situation meaning it's not possible for people to get that experience now in time). Graduate entry medicine is usually more competitive than undergraduate entry medicine, so as I understand most GEM applicants also apply to one or two standard entry medicine courses. However there are some significant financial ramifications of doing standard entry medicine as a graduate, so you will need to carefully assess whether that is a feasible option for you.