@mindutopia - I haven’t had private thyroid testing done (mine was covered), but as no one has answered your question, I went digging for answers for you. Here’s some information that I hope might by helpful.
First, someone asked the same question here in 2014 (just proves needs don’t change, do they!). Obviously, the information will be a bit dated, but here’s the link to that thread with some answers of private thyroid testing facilities in London, and one to avoid, it appears:
www.mumsnet.com/Talk/general_health/2188946-Private-thyroid-testing-in-London
Next, ThyroidUK have a page with information about private testing. Here it is:
www.thyroiduk.org.uk/tuk/testing/private_tests.html
Both they and the 2014 thread mention Blue Horizon, which will run the tests for you at a London patient centre. I read reviews and most Google reviews were five stars except for two (one complained about wait and another complained they weren’t diagnosed properly, and a rep for Blue Horizon politely explained they weren’t supposed to diagnose). Here’s a link to their London location: bloodtestslondon.com/
A full “London Thyroid Check” which includes TSH, T4, free T3, free T4, Thyroid peroxidase antibodies, Thyroglobulin antibodies, Vitamin D, Vitamin B12, Ferritin, Folate and C reactive Protein, costs £169.
I would imagine other places would probably be in the same range to stay competitive, so maybe between £150 and £300, depending on if they have a consultant explain the results to you and things like that.
Sorry I don’t have more specific information, and good luck, OP. I hope you find the answers you need!