Hi Bethan.
The quick answer to your predicament is that I see no reason for you to have to wait 3 months to take a thyroid test, unless you taking the sea kelp tablets has any bearing on that decision, which even if it does, three months is still a long time to wait.
I have hypothyroidism, and was only diagnosed as diabetic a few months ago. I also have a type of autoimmune arthritis, so I do think, although I can't be positive, that these conditions are connected in some way.
Having an underactive thyroid makes it much more difficult to lose weight, you feel constantly fatigued, where just everything is too much effort, and my hair at least has thinned quite considerably, although I do take other medication which may add to this.
You certainly seem to have many of the symptoms, so if I were you I would go back to your doctor and insist that you get a test done earlier. It's very simple, just a blood test which takes minutes, and from there, hopefully you will get either a diagnosis of thyroid problems, or perhaps another reason for feeling poorly, ie anaemia, which can mimic the same symptoms, but which is easily dealt with.
Unfortunately many doctors in the UK seem to refuse to put patients on a thyroid replacement drug until they reach a certain 'level' (Sorry, I can't remember the exact details offhand) whereas Doctors in the USA are commencing treatment at the lower end of the scale.
I wish you luck, and don't let the Doctor make any excuses. It's your body and you are the one who has to take care of it.