Hi! I'm in the same boat at the moment. Public (NHS) and private are the two streams available.
For the NHS, you need a GP to refer you for an adult screening. It has to be a GP, and make sure they don't fob you off on the wrong service, I got sent down a few rabbit holes before getting referred to the ADHD screening service. Then, you have to wait. I've been told that my screening should take 5-6 months to get booked in, and that's where I'm stalled at the moment. At/after your screening they will decide whether or not to book you on for a full diagnostic. The wait list for these is ~3 years. Yeah, very oversubscribed, super cool.
Given that I am looking to go private, which means that I need to start identifying private providers and what their charges are for screening, diagnosis and ongoing treatment. I've heard that these cost around £1000 for the diagnosis and then around £100 per month for any medication, but I haven't added these up myself yet.
I've heard that there is a route where you can ask your GP to refer you to a private practitioner they have an agreement with, which basically means they will accept a private diagnosis so you can get your meds (if required) on the NHS, where mostly the NHS is unlikely to let you get a subscription for meds without an NHS diagnosis. I've personally ruled this out because my local NHS have been no fun and I don't want them to piss me about for another month.
If anyone else knows of any options then I'm all ears, because right now it's either "don't get diagnosed for four years" or ££££ every year.