Hi Mike,
There are a few of us from the UK on the boards

Via the NHS the standard procedure would be first to let your GP know and for them to refer you to your local NHS trust's mental health team. You should be seen by a psychologist (i made the mistake of asking for a psychiatrist and that's a rather different field...) who will make a recommendation based on their assessment whether you should be referred onto one of the Gender clinics located around the country. I know this can sound quite daunting, but just be yourself and honest about how you feel and you should be fine. Once you are referred to a clinic, your first appointment would be with a clinician/psychologist who will run through what they would need from you in order for you to start Hormone replacement therapy. Essentially this entails changing your name via deed poll, and to start presenting unequivocally as female to society. It's part of what is known as Real Life Experience (RLE) and they would expect you to be presenting female in either a place of employment or in a social capacity.
Doing all this prior to attending a gender clinic will to a degree speed things up.
A few things to note, the NHS is not perfect. If your GP is anything like mine, she is absolutely lovely and really wants to help, but hasn't seen many, if any transgender patients walk through her door. This may also be the case with your local mental health team. The psychologist that you see, may not specialise in gender issues, and the priority really is to be passed on to someone who can better treat. I assume that assessment is to rule out any other potential conflicting mental health issues.
Wait times to see a gender clinic can be very, very long. I attend a trans group down here in London, and over the years a number of staff from Charing Cross Gender Clinic have come in to give talks and answer any questions, and they do appreciate that it can be incredibly frustrating waiting 4-6 months or longer potentially between appointments.
The RLE experience isn't great tbh. I personally feel it's downright ridiculous to force people to do this prior to receiving any HRT. I am sure other people will disagree, but i find informed consent a much better model.
There are other options.
1) Gender care is a private clinic based in London that effectively employs a resident psychologist and endocrinologist from Charing Cross gender clinic. They enable you to sidestep the initial wait times, by having two of your meetings on the same day and then being passed over to Charing Cross or your local NHS gender clinic for continued treatment (As opposed to waiting 4-6 months for your first appointment). Even though it is private, this doesn't mean that you can also sidestep RLE for hormone treatment. It is also quite expensive. (gendercare.co.uk)
2) The other option, is another clinic based in London, called Transhealth. The clinic is run by Dr Richard Curtis and having seen a gender therapist at the clinic for a minimum of four sessions, he is able to prescribe hormones. He will conduct an assessment and discuss reasons why you may or may not be full time at the time of HRT commencement, but being based on an informed consent model you do not have to be full time to start.
A few things to note, Dr Curtis and his team are absolutely lovely but as other have mentioned on these boards he is not an endocrinologist but a GP specialised in Gender treatment. He also absolutely disagrees with the prescribing of progesterone. Gender clinics in general disagree with his practice largely because it operate outside of NHS guidelines (i.e informed consent as opposed to RLE). (transhealth.co.uk)
I know this all can be quite daunting, but please don't let this stop you from thinking about transitioning. Please don't hesitate to get in contact with any more questions or for a chat