Email the national autistic society.
NAS
I went through the NAS - spoke to a really nice lady who wasn't judgmental at all and she told me exactly what I should do cause of the anxiety - I actually went to the place after they said I could pop in for a chat as it's local to me.
Basically you'll get a lot more respect off the NHS doctors and taken more seriously if you say the NAS referred you. Then you simply make an appointment with a GP but you ask them "Do you have a specific GP who deals with autism, or know who the best GP would be to see about that?" They will book you in and the GP will do an assessment test then refer you to hospital appointments with a specialist so they can test you and go through the past with you, like therapy kinda to get a better understanding to see where you could be at with autism. For me I had selective mutism in primary/high school and they kinda went off that and knew pretty easily from the experiences I told them about.
If you have money though you can just get the NAS to diagnose you, and they are far better than the NHS.
In my experience the doctors are much more kinder if you actually say you have really bad social anxiety, they will know how to deal with you in a better way if you just say you have strong bouts of social anxiety, there's nothing wrong with being scared.
But yeah - they were really nice with me and it was quite a fast process. Good luck and if you have any questions let me know.