First step is to speak to your GP, even if you're going private.
I went private with Dr Curtis and I would seriously not recommend him. I've had a lot of problems with him. (Check my "Accusation from Doctor" video for more info) He is the most well known trans specialist though, and definitely one of the quickest ways to get hormones. You'll have an hour's meeting with him for the first appointment and then he'll hand you the informed consent form. This means that 4 weeks later, you will be able to return and start testosterone assuming that your blood tests came back clear.
Dr Curtis is also very picky about the different forms of testosterone. He highly recommends T-Gel and Nebido and hates Sustanon, which is the standard form of T prescribed by the NHS. If you want Sustanon, you'll have a hard time convincing him to prescribe it.
There are other trans specialists, and I would recommend doing some research and maybe finding someone else. From what I hear, the others are cheaper too. (£90ish for appointments instead of Dr C's £230 for first appointment and £120 for follow ups).
Your GP should be able to sort you out with counselling with either the practice's counsellor, or the local mental health team. Before starting T, it's usually required that you have a mental health assessment to make sure that you don't have any mental health issues that maybe affecting your decision making or reasons for believing you're trans. It also helps the process if you've legally changed your name, have been living as male and have a letter from your GP.
Of course, if you're happy with waiting, then the NHS is completely free. Living in the SE, you should be able to get a referral to Charing Cross Gender Clinic in London which is by far the most reputable and reliable in the country. Problem is that you need to have lived as male for a year on the NHS before hormones will be prescribed. There's obviously a reason why that measure is in place though. You'll find far more "gate-keeping" on the NHS. They want to make sure that it is definitely the right decision for you. Whereas hormones are given out far more freely if you choose the private route.
Hope this helps.