Best to aim for an IBMS accredited course in that case. The ideal course is the Healthcare Sciences (Life Sciences) courses that are/were part of the NHS Practitioner Training Programme (PTP). A lot of these have closed in recent years as they transitioned to an apprenticeship format instead, but those courses are not only accredited by the IBMS but include placements to complete your professional portfolio to register with the HCPC upon graduation, allowing you to directly apply to Band 5 BMS roles after you graduate.
An IBMS accredited course which doesn't have these placements only gets you halfway to working as a BMS, as you still need the HCPC registration - for other courses you'd need to look at those with a sandwich year/year in industry type placement where there are possibilities of a placement in an approved NHS pathology lab (however you would need to apply to those competitively so a bit of a gamble), or you would need to work as an MLA for a few years after graduating to build your portfolio (with approval from your manager) before you can apply to BMS roles.
The apprenticeship format of the healthcare sciences courses are also worth a look, although a lot of these are now I think designed for internal NHS applicants currently working as an MLA in an NHS pathology lab.