Hi, and welcome!

If you're not yet able to undergo medical transition they don't need to know for medical reasons... so your main problem at this stage is most likely that they're going to call you 'Miss So-and-So' and misgender you all over the place. So it all depends on whether you want them to call you by a new name and pronouns... irrespective of whether you've legally changed it. If you're fine to go by your birth name for now, you don't have to do anything.
I went to the Receptionist at each practice and got them to change my details on their records. You can ask them to put a preferred name (or nickname) on your records for now if you like, so even if you don't have a new name yet, at least you won't be triggered by them using the wrong one.
Then when you see the doctor you can choose how much (or how little) you want them to know. If they say 'Oh, hi Jane!' and you say 'Actually, it's John now; I'm transitioning' that should be more than enough info. They'll probably ask whether you're on any new meds or anything, just so they can make sure whatever they're prescribing you will still be OK. Be aware that doctors are human and some may ask you inappropriate questions out of sheer curiosity, so feel free to decline those questions if you don't think they're relevant to your treatment. After all, a rheumatologist probably doesn't need to know about your privates.