Most trans people don't have a GRC here. A GRC is mostly a way to change your birth certificate and death certificate so if your birth certificate is already corrected then you are fine. Passports will let you change them with a doctors note from any doctor. HRT might be a bit of a pain depending on what part of the country though some GPs do offer bridging proscriptions while you wait to see the GIC.
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I can't comment on the immigration complications to it, but with regards to GRC and access to hormones, it's a right pain in the arse.
If you can afford the private fees, it's simpler. Though you will have to pay £100s for things like blood tests, and medication.
If you would like a broader view of costs, let me know.
I think a lot of people use private services as a stop gap while they wait for the wheels to turn/find a GP who isn't obstructive.
I'm not trans or an immigrant - so please feel free to ignore my thoughts on the matter.
Anyway, regardless, it's a good question, but probably not one with an easy answer - I think in this case, the best option might be to ask the people who make some of those decisions - in theory, if you look at the webpage gov.uk Apply for a Gender Recognition Certificate, there's a few email addresses and phone numbers that may be able to answer half of the question - though separately you may need to speak to the immigration/visa people gov.uk partner/spouse visa for the other half of the question. All in all, you've got a few layers of complexity here.
My own experience with contacting other government departments on different things is "very slow to respond, but then surprisingly helpful in the end".
There may be some useful advice on websites such as transactual.org.uk or transinformed.co.uk - your specific situation isn't likely to crop up in a FAQ, but you may be able to pick up half an answer to start with - then there are contact details on there which would hopefully lead to some useful information.
Sorry there's no clear and immediate answer I can find.
I wish you the best of luck!
Have ti say. Im sorta interested to findbout what the answers are.
I support the community. And am annoyed at how little recent governments have.
But info like this really only comes up when you know folks going through the process.
tldr. Thanks for posting an interesting question