My just turned 18 ‘Non-Binary (probably), for now Just Me’ identifying child recently had a similar experience with their gp. Lots of confused looking, shrugging, sighing indifference and a reluctant referral to the GIC once we explained that it even existed.
Weeks later, said teen child received a letter from the GIC asking for another very basic form to be filled in (which made me question whether the gp had filled the original one incorrectly) and stating that there would be at least a years wait (as you say). End of correspondence. No further information, no further contact — as far as I can see, no further evidence of anyone giving a damn about the health and wellbeing of this young person who has reached out for help.
It’s almost as though there is a secret consensus that if a person is indifferent and uninterested in choosing a polarisation of the gender binary, then the government/NHS/society at large has a right to be indifferent and uninterested in them.