Doctors in the UK are less likely to do it. For one: They often outsource their phlembology to a local hospital rather than do it themselves now. For another: in the US if you give a doctor money and what you want is not unethical or against the law, they'll do it for you because you are paying them. In the UK there are no private GPs (that I know of) only private specialists who you have to be referred to by an NHS GP.

Had I still had the job I used to have, it would not have been a problem. I could have invited you in to the lab, got a doctor (we had some very good medics who were very good at taking blood) to take a sample for my research, put it into some anticoagulant and, if a small amount was sidelined into another vial, no one would worry about it so long as you were consenting to that sample being taken and used as it was used...