You're in luck. I have the names of three doctors in the London area. All have been recommended by fellow Forum members. Sending PM with names.
I'm in London thanks for your help
Many doctors won't treat subclinical hypothyroidism (elevated TSH, free T4 in range) and the travesty of this is you have to suffer for a lot longer (could be years) until your are "officially" hypothyroid (elevated TSH and low free T4). Iron, vitamin D and vitamin B12 are common deficiency states found with hypothyroidism.
Check thyroid antibodies to see if have Hashimoto's thyroiditis: thyroid peroxidase antibodies (TPOAb) and thyroglobulin antibodies (TgAb). Positive results could convince a doctor to start treatment.
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"There is also the difficulty of interpreting a serum of concentration of TSH in isolation. A concentration at or near the upper limit of the reference range, particularly if associated with a normal T4, may indicate autoimmune thyroid disease. A consensus exists for early treatment of such patients with thyroxine if anti thyroid peroxidase antibodies are present in the serum, because the risk of overt thyroid failure in future years is high, and it makes sense to anticipate morbidity rather than risk loss of follow up."
Drs Anthony D Toft and Geoffrey J Beckett, "Thyroid function tests and hypothyroidism", British Medical Journal, 8 February 2003, 326:296-7.
Unfortunately it seems that most doctors in the UK follow the NHS guidelines and only want to diagnose and treat based on TSH levels. That simply doesn't work. Scientific studies have shown that Free T3 correlated best with hypo symptoms, while Free T4 and TSH did not correlate at all. Some of our members have been successful in getting the testing and treatment they needed. This is a quote from such a member in the UK.
"What I have learned from my experience is that you have to go to the Dr's office and TELL THEM WHAT YOU WANT and to go backed up with knowledge. You have to tell them that you have done your reading and looked into your condition and care about the long-term treatment of your health and thyroid. If you fight for what you want, you will eventually find someone that is happy to go along with your wishes. But we all have to take charge of our own health, right?"
So I suggest that you should learn all you can about hypothyroidism and be prepared to refute what the doctor tells you, and push hard to get the testing and treatment you need. The only other options are to look around for a different doctor that will treat you clinically by testing and adjusting Free T3 and Free T4 as necessary to relieve symptoms, or to go private.
I do know of a few doctors in the UK. Where are you located?