The term "biomedical scientist" is protected in the UK and so the only way to be a biomedical scientist is to have an IBMS accredite biomedical sciences or healthcare sciences (life sciences) degree and to be registered with the HCPC after completing a professional portfolio while working in an approved NHS pathology lab. That process is then somewhat mutually exclusive with being a doctor, unless you then do medicine as a graduate. However, you wouldn't really be a biomedical scientist anymore at that point as you wouldn't be in s post with the NHS as a biomedical scientist.
To be a research scientist, in medical sciences or other bioscience fields, would generally imply you were an academic at a university, which would require normally that you completed a PhD and psordc and established yourself as an academic. There are routes in medical training to support a career in academia while training as a clinician.
To be a biomedical engineer would mean to work as an engineer. You couldn't work as an engineer while also being a doctor, the need for multiple separate and unrelated degrees aside, because they're totally different fields, even biomedical engineering. BMEs might work with clinicians at various stages but the roles are distinct and non-overlapping. This is also true of the biomedical scientist role as above.
Essentially there is a lot more to being a (biomedical) scientist or engineer beyond merely getting a degree in the area and it is fundamentally a full time profession which is separate from being a doctor. So just as you couldn't realistically simultaneously work as a doctor and a lawyer, you can't "be" both a doctor and a biomedical engineer or biomedical scientist (in the sense of the meaning as a protected term), although you could be a doctor and a researcher doing scientific research. You could get degrees in those areas before, after, or while getting a medical degree, but that by itself wouldn't make you a scientist or engineer as indicated.