So, sleep_supremacy is mostly right. A biomedical science degree doesn't make you a biomedical scientist (odd, but true). Biomedical scientist is a protected title in the UK. It means that calling yourself a biomedical scientist and/or doing the job without being on the Health and Care Professions Council (HCPC) register is against the law.
To get HCPC registration as a biomedical scientist you need a certificate of competence from the IBMS. Which means you need an IBMS accredited degree or a degree they have signed off on. I've never (in almost 20 years) seen a case where the IBMS signs off on a non-accredited degree without top-up modules.
If you think (or know) that you want to be a biomedical scientist as a profession, then switching now seems more sensible. If you think you'll want to go into research / graduate entry schemes / industry etc, then it's not an issue.
Top-up modules is a relatively straight forward process - you apply to the IBMS, map your degree against their requirements, and then do the modules they require to get accreditation. However, you then have to study in your own time, and pay for it when student loans are no longer available. Looking at the University of Westminster, it's currently £1500/module which is a lot.
Depending on your current course, you may not have to do the whole first year again. I'd look at what they call 'accreditation of prior learning' and you may need to spend 3 years, but not do it full time in the first one. (Then I would recommend finding your local hospital and applying to their staff bank as a medical laboratory assistant so you can get work experience and be paid for it.)