Thanks for the link, I had not looked there. Thanks also for pointing out that a private diagnosis may not always be accepted in order to access services - I am already under the disability service in education for mental illness and I have heard there isn't much NHS support for adults with autism, so if I got an ASD assessment it would be more for me to know if that's where my problems are rooted or not, rather than to access services.
Unless I have made an error searching, that site seems to say that the only diagnostic service for adults in the UK is the Lorna Wing Centre for Autism in London. I would be willing to travel to London, however it says that they only accept NHS referrals there. Every time I go to the GP to ask to see a specialist, they tell me they will put me on waiting list, but it always turns out that the waiting list leads back to a psychiatrist, and the psychiatrists always seem to have no idea about autism; when I ask about it they kind of just shrug or raise their eyebrows and avoid the question by stating simply that I'm just mentally ill, so I go round in circles.
I think I will email the national autistic society though to check if they know of any private services that haven't been listed on the site.