Private Asberger Diagnosis? UK

So I'm 21, still in Uni whilst running a website business and working part time to support myself.

I have the option to go for an assessment with a private psychiatrist since my local GP isn't accepting any new appointments every time I've called up for the past few weeks.

I've been passed through the NHS to therapy for OCD/Anxiety/Depression multiple times in the past and it's taken months to go through the process each time. I'm less reliant on seeking therapy or medication (Never have tried it, and am uneasy at the thought) as I have everything (Bar my social life) fairly well planned out.

I've had people suggest Asbergers or autism before might be the reason for my symptoms, and once it was said to me it made sense, and the things I've tried to treat in the past (OCD/Anxiety) just kind of made sense. Obviously I knew that wasn't a reason to jump to self diagnose so I called up a private hospital and they referred me to a psychiatrist. The psychiatrists office said they believed they could help and I was a valid possible assessment.

I've heard that private diagnosis' are often not recognised (I'm not planning on collecting any benefits or anything like that). If I do have Asbergers I would like a official, recognised diagnosis just for my own peace of mind. Also to explain to people, my GP or other medical professionals should I ever need to. I'm worried the NHS or anybody really might say 'It's not a real diagnosis, or you need a second diagnosis'.

Would going the Private diagnosis route still be recognised by the NHS and major organisations?