Regarding T I find it better for the doctor to know everything. The reason for this is because if they don't know, they assume you're female and will either question you about your use of T (and likely the dose), or they might alter prescriptions so that your dose is lowered even without your permission. This happened to me recently, when using the local surgery's repeat prescription service. I put my repeater in the box and some doctor signed off on it who doesn't know me and doesn't normally see me. When I went to pick up the prescription I found the pharmacy had bagged a half dose for me and the dosage had been reduced to over half my usual. Better for them to know or you'll have to go through the rigmarole of getting it put right like I did.
So yes, not being known by one doctor meant they screwed with my prescription thinking they were doing the right thing and obviously hadn't looked at my records properly, and that affected the service I was getting, so to speak.
As for the question of "will they know" - if your local surgery is set up with access to a computerized medical record system (which in the UK where I am is all of them) then it's all right there for them to see, including the history of referral to the GIC. GPs typically consult your records every time you go in to see them about something. Nurses also do, in my experience (i.e. for blood draws). Not that I've ever had a bad experience with any of the doctors or nurses to date. They have all been courteous, friendly and professional so far.
A dentist might not know, nor might a specialist in a hospital of some kind, unless they want to access your records. If you're going for major surgery I would expect they would consult your records, or try to. It would be haphazard of them to operate on someone knowing nothing about their medical history, I imagine. But obviously if you go for some unrelated treatment, there's no major reason to disclose your entire history. In the last two weeks I went for some emergency dental work in Liskeard and they gave me a questionnaire in the waiting room, the purpose of which is to ask if you are taking any additional medications so they know what may clash with some of the medications they were going to be injecting into me. Since T is the only medication I am on, I put it on the questionnaire. Nothing was said or asked about it when I went in for the treatment so I assume it was irrelevant in that instance. I would always advise disclosing all your medications if you're going to be operated on, or if they are giving you other medications so they can warn you or make adjustments if your hormones are likely to clash with something. It's for your own good.
When visiting the doctor for something as routine as a sore throat prescription or some other checkup completely unrelated to being trans, I wouldn't see it as necessary to go into it my history with the GIC or hormone replacement.
I see the same doctor each time at the GP surgery which I find is worth taking the time to arrange because she knows my history with the GIC and my ideal T dosage that we have worked out so far, etc. Other doctors might need this describing to them so I tend to try to make appointments with her if I can. At the GIC, though, I have seen many different people and had to explain my situation more times than I can remember, which is exhausting. If you can snag a good GP or therapist, try to keep them. It makes things a lot easier.
That said, now that I've changed my name and sex, requested to be taken off the corporate sharing NHS database in the UK (not the medical one but the one that allows certain bodies to look up your details who are not doctors) I've been assigned a new NHS number and the gender allocation of male... and whether that will mean my old history is wiped completely or retained I have yet to find out.