Where to go for autism diagnosis?

I live near Edinburgh and am currently looking for a place where I can get diagnosed for autism.

I was on the NHS list for the assessment and had my first consultation with a psychiatrist recently. As he's not an autism specialist, he said he will refer me to a psychologist from the same hospital.

This means that I'm now on another waiting list and it can take them nearly another year to offer me the appointment with the NHS psychologist.

Is there any way I can get diagnosed privately? What are my options?

Many thanks in advance.