Hi Alice,
I'm not sure which private blood test service you've found online, but most offer a venous collection option as well as the home finger-prick test. You normally get the chance to select this during the checkout process. Obviously, this will cost more, but they'll use a number of private clinics or hospitals nationwide so hopefully you'll have one fairly close to you if that's the option you choose.
I never had a baseline test done, but I'm prescribed hormones privately by Gender GP and after 3 months I asked my GP for a blood test to check my testosterone and oestradiol levels stating only that I was receiving hormone treatment privately. They arranged the test and didn't really ask too many questions (although the receptionist was a little surprised by the results when I went in to collect them and I had to explain that they were actually good). Perhaps your GP would agree to a test if you said you were starting hormones, which you were obtaining privately?
When I went for my 6-month blood test a couple of months ago, I asked for a referral to a GIC and I've just been called back into the surgery for the blood tests requested by the GIC. They haven't requested hormone levels; only cholesterol, diabetes, liver, kidney, urea and full blood count. So although the wait for the GIC may be long, if your region is anything like mine you may be offered blood tests sooner, once referred. I certainly wouldn't consider obtaining these tests privately and they're probably the ones you had at your recent health test anyway.
I hope things work out for you!
Casady