My experience may be way different from what you'll go through, depending on the doctor. I go to an extremely trans-friendly clinic and they anticipate there being a lot of dysphoria throughout the process, so they've modified it from a standard physical. I also go to an informed consent clinic, so that part of the process is a little different. That said, this is what they did:
- Came into the exam area
- Got vitals taken (blood pressure, height/weight, temperature, etc.)
- Waited for doctor
- Discussed medical history (mine & family's)
- Got recommendations based on the above (her recommendations: flu vaccine, HPV vaccine, pelvic exam/pap smear - I declined consent for all of them)
- Went over the informed consent form
- Discussed types of T, my preferences, and her recommendations
- She listened to my heart, lungs, and felt my lymph nodes and abdomen
- She put in the orders for my lab work and sent me down the hall
- Scheduled my next visit
- Got blood drawn
- Drank water and ate a snack before leaving
It was very non-invasive and the environment in general was very comfortable. My chest was never touched aside from having a stethoscope on it for maybe 15 seconds. I never had to take any clothing off.
T injection sites are in a few specific places. If you're primarily going to be self-injecting, I'd say you'd prefer doing them in the thigh and see if you can leave it at that. Perhaps also wear or bring with you underclothes (not underwear, but an undershirt and maybe some long underwear) and see if you can't wear those instead of stripping down if they ask you to. I know you said NHS so you're in the UK? Not sure what the attitude there is, but at least over here there wouldn't be any issues if I brought a friend/parent/significant other for support. Maybe consider having someone go with you to help you relax.