Lola Health have possibly the best range of testing offers which include visiting phlebotomists. For some people, this option makes the difference between being able to get a test done - and not.
Personal experience showed that they are efficient and the doctors' interpretations were properly considered and helpful. Obviously this is on a very small number of tests.
(You can avoid using the referral link above, if you wish: Lola Health without referral.)
Thyroid UK lists several private testing labs. However, there are other private testing labs which are NOT listed but might be particularly helpful to some people.