I don't know if this works at other GP practices, but am sharing in case it does. As I (accidentally) managed to get my full thyroid test done on the NSH (hopefully), now just waiting for results.
I went to the GP and asked for full thyroid test. He said NHS doesn't do that. I explained why I need it, GP wasn't interested, said he didn't believe that there was anything wrong with me, blabla bla..
He was a student doctor and finally he agreed to discuss it with his mentor.
On the day of the blood test, I find out he hasn't discussed it with his mentor. Turns out he just wanted to get me out of his office!
I'm called in to the nurse and I ask her to check which blood tests I'm down for. And I explain that the doctor was supposed to check with his mentor and I should be getting a full thyroid test. She has a look in the computer and asks which ones and just clicks T3 and T4 on her screen!!
I couldn't believe how easily she could do that. Mind you, she is a really good and experienced nurse. But I've learned it might be worth asking the nurse who's taking bloods to send it to the lab for a full thyroid test.