Amanda, mixing private treatment with NHS treatment does not go down too well in NHS gender clinics ... they're likely to say "well make up your mind who's care you want to be under" .... "if you can pay what are you doing here ?".
If you have had private sessions, and are receiving meds on private script its likely to act as a "black mark" against you (of course you don't have to tell the NHS GIC - though you might have to explain your source of HRT which blood tests will show up).
If you want the NHS to pick up the tab for GRS then the only option is to co-operate fully with whichever GIC you fall under .... contrary to belief they're all reasonable these days, problems are invariably with Local Area Health Authority / PCT funding policy which varies so widely across the UK.
Most GICs will prescribe HRT on second or third visit (within 3-6 months, as per Benjamin guidelines) if they "diagnose" you as TS, especially if you've already started RLE.
You don't say where you are in the UK, but if you are in the majority who are sent to London CHX, then they only accept referrals from a registered NHS psych, not directly from a GP .... so there are two waiting lists, one for psychotherapy and however many sessions until you're given that referral, and then the wait to be seen by the GIC.
You can self refer to CHX NHS GIC on a private basis with the "introduction" being via your GP (£200 for the first appointment was what I was offered - same as going private) but then I guess you have to pay your way, except that your GP may be "easier" with converting their scripts to subsidised NHS prescriptions, and maybe NHS funded GRS would remain a possibility.
If you go the private route like I did (would be Dr Richard Curtis these days if you're near London) then its £200 first session, £100 every 3 months for subsequent ones .... a year RLE to get the "letter" .... but it sounds like your GP might re-write the private scripts to subsidised NHS - mine didn't although he was willing to refer me to a TS experienced NHS endo for full blood screening.
Private HRT is £750 per year, internet prices £1800, NHS "season ticket" I think about £120 .... GRS is £10,000 in UK/America, half that in Thailand .... if you know what you want and can afford £8 -12,000 it will save you two years of waiting and give you control over your choices.
What price do you put on each extra year you gain as your "true self" ?
Laura x