They make it so easy. NHS waiting lists can be years long. I was on a "9 month" waiting list for nearly a year. Since following the instructions here it has been 2 and a half weeks and my first appointment is about to be booked.
As long as your GP isn't an ass and accepts that your self-assessment form and any evidence you have is enough to justify an assessment, you will be able to get an appointment. You just give forms/letters to your GP and Psychiatry UK do the rest - even chasing payments from your CCG.
In my experience, GPs often aren't familiar with how Right to Choose works or confuse it with funding requests, which the CCG will likely decline. There is a letter you can print on the page I linked that outlines Right to Choose for the GP. I also edited mine to clear up that it wasn't a funding request and didn't require the same protocols.
Once your GP has sent your referral to P-UK, they will contact you. You then sign up to their portal where they will send you forms to be filled in. 3 are for you and one is an "informant form" for someone who knows you well. This is the only real difficult part because you might not have someone who knows you well enough/believes that you have it/ you could be like me and know your parents would just dismiss whatever you have to say about it.
Fortunately for me my gf is supportive and was willing to do it, I also asked P-UK if that was appropriate, they said "The Informant Report is ideally completed by someone who knows you well and has for some time." It's probably fine for say, a close friend who has known you for at least a few years and at least somewhat understands ADHD and knows you want to get diagnosed. (Also it's very funny comparing what they notice about your symptoms to what you notice)
For some reason in the UK, prescribing for ADHD is like pulling teeth, and when doctors are willing to prescribe it's very rarely amphetamine. Compare the rates of prescription makeup from America to the UK. What's troubling there is the fact that atomoxetine is prescribed at a superior rate to amphetamine, which is absolutely not supported by evidence. It pretty clearly underlines that bias and stigma still structure attitudes towards treatment.
On the bright side, methylphenidate has good evidence in support of it still and perhaps the highest tolerability, but consider pushing for alternatives if they stick you with atomoxetine and it doesn't work (although give that one two months) or methylphenidate has troubling side effects.