You should be able to work in industry without HCPC registration, I believe; you just can't work in NHS roles called "biomedical scientist" (which I think is a protected title, so searching for "biomedical scientist jobs UK" would only come back with those kinds of roles if so). You may be better off looking for slightly more generically named positions like "lab manager", "research technician", "lab technician" and so on, if not just more widely at other grad schemes which may or may not be in scientific or para-scientific areas.
If you specifically want to work in a bioscientific lab environment, that isn't necessarily in the NHS (which would require the HCPC reg and IBMS accreditation) you should also start looking at PhDs, because that's what those kinds of environments need generally, postdocs and established researchers. Also look at university bioscience department job offerings, as often they'll hire grads to act as techs and those roles may involve more non-trivial work under a researcher, which might at least be decent paying work while you look for other positions or apply to PhDs.