Is your adhd referral on the right to choose pathway is my first question? As that can be faster.
Regards to going private, pre 2020 it may have been the case that your GP might then take over prescribing under shared care, I don't think that is now the case as they see it as jumping the queue. But it may be worth double checking with your GP just in case, let them know you came into some money that could fund the assessment. It would not affect your existing referral to check.
You need a psychiatrist IMHO. I don't think a GP with expertise in ADHD can actually prescribe ADHD meds. Only a psychiatrist can do that. So they might diagnose but then you have to go somewhere else for meds who might not accept your diagnosis and want to do their own assessment.
I used Psychiatry UK. It involved assessment forms, one assessment appointment. We then had some less expensive follow up appointments while I titrated on medication. Once you are titrated on medication you must have and pay for an annual review as a minimum I believe. Mine is £250. If you go private and the NHS will not take you on as shared care (as I said, less likely these days) you will have to continue to pay for the cost of your prescription (c£15) and monthly meds (anything from £30-£70 a month I reckon). If the NHS accepted shared care the GP prescribes so you just pay your usual prescription charge and not the meds. A GP will not deal with any changes or tweaks to your prescription, so if any issues you have to go back and make an appointment with the psychiatrist. So its very important to take your time during titration and not rush it to get it right.
If the NHS won't accept shared care, you will need to go through the whole assessment and titration process again when you reach the top of the NHS waiting list, if you don't want to continue to pay for meds.
So essentially, it's only really worth going private if it's meds you really think you want/need.
Otherwise you have a paper diagnosis - it might help for work purposes as its a disability under the equality act so protected. But you could always inform an employer you are awaiting assessment. ACAS are helpful on work issues and give free advice. And you can use Access to Work even if you don't have a diagnosis (approximately 7+ months wait last time I looked). You can read lots of books and podcasts on ADHD too, as chances are if you think you have it, you probably do (if you know, you know!)