Thank you all SO MUCH for your support!
I'm reading this thread on my lunch break and really glad no-one else is in the office because I'm getting all teary eyed.
After more cancellations and faffing around with GP practice, I finally saw a doctor yesterday to discuss. My TSH from blood test last Monday was 5.27, so even higher than before despite taking a higher dose. GP have now increased my dose to 50mg per day and I will retest again after 8 weeks.
The GP I saw this time was fantastic, I told her everything and showed her my list of tests, she agreed that all of those are reasonable and I'm going back on Thurs to have those done. She also added on a few tests to rule out any other conditions (specifically mentioned diabetes), and to test kidney and liver function.
She asked me to book in with her again once these blood tests are done to discuss this plus depression. She said that although it is a symptom of hypothyroidism she wants to make sure they are not just treating thyroid and ignoring depression.
I was a bit overwhelmed & tearful so forgot to ask for a print out of all my results, but I can get this when I go back on Thursday.
IvyKaty - I'm 34. I had hypothyroidism and endometriosis symptoms for approx 3 years and kept going back to my old GP surgery only to be told gems such as "all women have painful periods" or "everyone feels tired and gains weight as they get older" or (best yet) "I have other patients who are worse off than you that don't ask to see specialists".
The day I got my hypothyroidism diagnosis (which was only because I saw a locum who was shocked I hadn't had blood tests and ordered some), I went straight to register with my current GP's practice!
Yellow - thank you for all your advice. I've just ordered the book 
It took me a long time to admit to myself that I'm actually unwell, not helped by old GP attitude & to lesser extent family who are a bit like "but you've got pills now, just take those and it'll be sorted out". My OH tries to be supportive but also has trouble understanding exactly how bad I feel now. I explained how depressed I am, partly due to weight gain, and his answer was for us to sign up for a Tough Mudder style race next year! I know he is just trying to help but this just feels like added pressure to not only get better but to get to some elite form of fitness. I am going to discuss with him soon, when I'm feeling a bit more resilient and not liable to keep bursting into tears.
I spoke to my boss yesterday and explained how hard I'm finding things, she was brilliant and has agreed to let me do 1 day a week working from home (I'm going to do Weds from home to help with mid-week exhaustion). I think that just being able to avoid my usual commute and spend the day working in my pjs will be really helpful.
Bobby & Kai - I think I will push for private endocrinologist referral in the new year. I'm lucky enough to have BUPA cover through work (although not sure what level cover we have), I will do some investigating this week on coverage. In meantime, I'll look at supplements etc.