I was referred via Right to Choose (which I believe has got very busy recently due to it becoming more well known but is still very quick compared to the NHS waiting lists) to Psychiatry UK. Basically under the NHS constitution you have the right to choose your mental health provider - that's the gist anyway - and P-UK are an approved provider, regardless of where you live in the UK. So you are referred by your GP and your CCG pays for it. Back when I was diagnosed a year ago it took a week from referral to starting medication, but I think it's about 4 months all in now - still much quicker than my local NHS waiting list of 2.5 years.
So I was worried about them being unwilling to refer me for ADHD and actually, the doctor immediately agreed I merited referral, it was explaining the Right to Choose referral route that I got pushback on (which is common).
Would definitely check out P-UK's website and their page on RTC if it's an ADHD assessment you're after. (Also, @Thingsdogetbetter - ask your GP to change your referral and don't take no for an answer - it's your legal right and will be quicker!).
If you want an autism referral near me it's also a very long waiting list I think, but I'm not sure.
I was pleasantly surprised how open to referring for adult ADHD (I didn't have a child diagnosis for various reasons) my doctor was, but having spoken to others who got adult diagnoses that varies massively. It helps if you can find one of the official diagnostic self-report scales they use ahead of the appointment and fill that in, plus go armed with info and examples of how it affects your life. My GP does online appointment booking where you can attach files, so I attached that to give them chance to look over it before my (virtual) appointment, that helped set the scene for them so I wasn't talking over myself and forgetting what I wanted to say.