I have the same kind of problem. I was lucky enough to find a specialist in adult ASDs near where I live (Northwest England), and am currently seeing her for counselling. She's given me an unofficial diagnosis - that is, it's her opinion that I definitely have AS - and I'm currently trying to decide whether to have an official private diagnostic assessment with her. I probably will, as there are various reasons I'd prefer to do it privately, including the stress of having to go through multiple appointments with multiple people in different places (I am socially phobic and struggle to talk to strangers), having to talk through the issue with my GP, and the likelihood of long waiting times. I am able to pay for a private diagnosis. I've seen a few people mention that NHS support is withheld when private services have been involved, but I'm not entirely sure what this mysterious NHS support is - for example, the specialist I am seeing mentioned that employers are required to make allowances for someone who is diagnosed with an ASD and whether or not the diagnosis is private makes no difference to that.
You are welcome to PM me if you would like the details of the person I am seeing - if you live in the region of the north/northwest of England she is very good and willing to travel to clients, with additional travel costs of course. However, if you don't live in that area the travel costs would probably be prohibitively expensive and you'd be better looking for someone closer to you.