I will say primarily, the NHS requires IBMS accredited courses because it focuses on the diagnostic side of things, as well as some ethical or legal side of things (including patient's opinions) such as the data protection act, human tissue act and consent to treatment and when is consent not needed i.e. the patient is passed out on the floor and bleeding out but you do now know if they have a religious or ethical issue with blood transfusions, this prevents lawsuits.
The generic consensus if if you wish to work in a NHS lab, you must have an IBMS accredited course. You can do a non-accredited course and then do top up modules but this will have to be done at your own expense and time. I once had a colleague who wished to work in a lab but, having done a non-accredited course, they would need to take up an additional 7 modules which would take 3 years to complete (may as well do a whole new degree in that time). I would suggest if you wish to work in the NHS, pick up an accredited course as, if you don't have an accredited course, you will not meet the job requirements in the NHS.
In order to meet the standards to work in the NHS as a Band 5 or higher (it does not necessarily have to be a medical lab as even paramedics fall under the HCPC but I am focusing on a medical laboratory perspective for this perspective), you would need to be a member of the HCPC (Health Care Professions Council) which can only be done by completing your IBMS accredited degree and then completing your IBMS portfolio in a UKAS accredited lab. This is typically done as a MLA or an AP but can also be done as a trainee BMS but these are HIGHLY competitive (I have applied for dozens of these and only succeeded in obtaining two interviews).
There are other jobs available such as clinical scientists but these also require IBMS accredited courses (or topped up courses) and require you to complete the NHS STP programme but again, this is highly competitive.
TL;dr, I think from my understanding (I'm not an authority on this but I have some experience), most jobs will require some form of an IBMS accredited degree and for you to be registered under the HCPC. For a full list of HCPC registered jobs, please click below and select "choose a profession".
https://www.hcpc-uk.org/education/approved-programmes/EDIT:
Changed the word prefers to requires.
EDIT2:
Having read what I did, I edited causing it to kind of not make sense or be vague (mix of lab and non lab jobs). Fixed now.