No, doctors don't test because they don't think T3 is important.
It would be nice if all doctors tested for T3 every time labs are drawn, but they do not. I think a realistic expectation would be that if patients are still complaining of symptoms related to a thyroid imbalance, doctors should order Free T3 tests to see how much of the hormone is in the body. I feel MUCH better with some additional T3 added to my T4 -- however, I do believe I could feel even better. It's a struggle & the bottom line is it depends on HOW YOU FEEL. If you are feeling good, I would move on and leave it alone unless I started feeling bad. On the other hand, if I feel down, depressed, tired, lethargic, cold, foggy in my head, anxious, etc., I push to have all of my thyroid hormones checked & treated.
The FT3 levels are very important, that's the one your body uses on a cellular level and if you don't have enough you will feel awful.
So as Zekat said, if you feel fine your level may be o.k. but if not then you want to have it checked because many times the problem of not feeling well is a low FT3.
But....the FT4 is also important and its very important to keep it in the upper 1/3 of the labs range, if not then you most likely will not have enough T3 either.
The UK is a very tricky place because of the medical council there (people forget that you are under socialized medicine). My relatives are all british, so I'm a bit familiar. You may have a very hard time finding a doc to even run the t3 free there.
I suggest you do some good Googling around the web for doctors who are thryoid renegades/mavericks in England. Perhaps that will get you going in the right direction. I do know there happens to be a good one in Birmingham.
That T3 Free is vital to your health if you ask me. I was near dead without it. Now I'm a big fan of Armour/Synthetic combo.
Just thought that I'd let you know that I live in Hampshire and am having no problem in getting the FT3 test done. Initially my Dr didn't know that the Free tests were necessary but I am lucky that she is forward thinking and called the specialist to check on what she needed to do.
I can only mention that it is very useful information. You may want to find some info about the usefullness of the test and print if off for your doc. Or...you can doctor- shop looking for a holistic type doc. Or...you can try to get it run by yourself. I'm not sure how that works in the UK... sorry about that.
Googling for that infor may be your best bet.
I know its hard to weigh the options...so I feel for you. I do wish you luck my friend.