Hi Ruruko, and welcome!
The NHS has recently changed its rules (in August) so you won't need to see a psychiatrist. Hurrah!
You say you're in the UK but there are different rules depending on which country you're in. I'm going to presume it's England and answer accordingly.
If you're in a rush to start HRT, you'll need to go to one of the private gender clinics. There are a couple of good ones in London, but they're extremely thin on the ground outside the Capital. You'll also need to pay £200-300 per session for their services. They can write to your GP asking them to prescribe HRT for you, and they can also give you recommendations for good surgeons if you want some procedures done privately. All things being equal, you could be on HRT within a few months.
If you want to go the NHS route, your first step is to see your GP. It doesn't matter whether they're your regular doctor; just go to see them anyway. Tell them that you have concerns about your gender identity and ask for a referral to a Gender Identity Clinic (GIC). They may ask you a few questions about your life experiences, childhood, sexual practices (yes, it is inappropriate but some of them do ask) but they'll write to an appropriate GIC for you. Your doctor is not permitted to refuse to send you for treatment; it is against the law to discriminate against a patient due to their gender identity.
Now... here's the bit where you need to be sitting down. All the NHS GICs in the UK are heavily oversubscribed at present so you shouldn't expect them to come back to you quickly. In most cases it takes several months... or even a year or two... to get your first appointment through. You'll then have three assessments with a Gender Therapist who will determine a) whether you have Gender Dysphoria; and b) whether you have any mental health issues that need to be worked on. After they've diagnosed you, you'll be sent on to the clinic's doctor who will prescribe your HRT.
The time it takes to get all this done depends on the clinic, but in most cases you can expect to wait between 1-3 years before starting HRT. Some clinics are quicker than others, and you are entitled to attend any GIC in the UK; it doesn't have to be the closest one.
There is another way, however... if you're in a rush to start HRT, you can (with your GP's agreement) ask for 'bridging' endocrine treatment, which is where your GP puts you on HRT while you wait for your GIC referral to come through. But your GP
can refuse this because they're legally responsible for anything that might go wrong.
Before contacting your GP, I recommend you familiarise yourself with the NHS Protocol for the treatment of Gender Dysphoria... and do yourself a favour: print it out & take it in to your GP, because trans* patients are rare so your GP might not be aware of the protocols.
Here is England's protocol:
http://www.cliniq.org.uk/sitebuildercontent/sitebuilderfiles/nhsenglandinterimgenderprotocolcpag12072013.pdfThe info about 'bridging' treatments is on page 15. The new protocol encourages the NHS to pay for facial hair reduction (see p 22) and breast augmentation & FFS are available on a case-by-case basis (p 12)
I also recommend you read this document, which is aimed at telling doctors how to treat trans* patients:
http://www.nhs.uk/Livewell/Transhealth/Documents/gender-dysphoria-guide-for-gps-and-other-health-care-staff.pdf This can be very helpful if your doctor doesn't have any other trans* patients and doesn't know what to do.
Flan mentioned about the Deed Poll: it depends on the clinic, but some clinicians will count the start of your RLE as being the date you changed your name for all aspects of your life. Mine did, which is the only reason why I'm on HRT in less than a year. But most GICs will count the start of your RLE as being the date of your first appointment.
Good luck!