Although I was a private patient, and a long time ago, I do have some insight into this because one of my very good personal friends (someone whom I holiday with from time to time) used to work in a UK GIC. He was a gender psychiatrist but is now retired. So I've heard from him how the system functions (or doesn't).
Firstly it isn't strictly true that GIC's won't accept GP referals - however in an attempt to save money some PCT's insist on your seeing a general psychiatrist first, so that they can filter out people who have other conditions and are nor genuinely transgender.
Secondly as to what to expect... it's a lottery, the result of which seem to depend on which doctor you get to see at the GIC, where you live (and therefore what financial constraints your local PCT has), and sometimes even what time of year you reach certain stages at - for example some PCT's regularly run out of money in February and so start restricting and delaying SRS cases to try and balance their budgets... the people then appeal - and get SRS on appeal - but the delay moves the expenditure on by six months!
So it is a bit hit and miss, but unless you can afford private treatment it's the best that you can do. It is also often painfully slow.
You will probably find that things progress like this... One or two appointments will be taken up with talking about your desires. No offers of hormones will be made at this point...
After two or more appointments they will have explained that in order to qualify for surgery you would need to do a real life test and if yu wish to pursue this they will start helping you to prepare for this step.
By the third or fourth appointement (by now you will have been attending the clinic for anything up to eighteen months) they will probably offer hormones if everything else is ok.
That is the point when many people start their real life test, which has to be for at least a year, but is sometime two years.
If you get through all that - by now anything from three to five years will have elapsed from the point where you started, then the final step is SRS.
There is no doubt that as a private patient you have more control over things and can progress in a quicker and more predictable way.
I hope all that helps... But remember - like I say, this isn't rocket science. Some people do get lucky even on the NHS and have a smooth two years from start to finish - but I would guess that they are in the fortunate minority. So I would prepare yourself to learn a lot about patience.