NHS or Private assessment for Autism

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I have a 12 year old son who I have always felt may be autistic but I have never been taken seriously by our GP or his school.
When he started high school in September he had a really tough time. Some bullying and stress resulted in a bit of a crisis which then resulted in the school insisting I take him to see the CAMHS crisis team. They said (amongst other things) that they felt he was likely autistic and sent out the screening questionnaires which myself and the school filled in and returned at the end of October.
Since then I have been waiting to hear from CAMHS to see if they feel he qualifies to be assessed. I have chased them up several times but apparently they are still waiting for a meeting where this is discussed. So 6 months so far just to see if he qualifies. We were advised it would be a couple of years once they decided if he qualified.
Thankfully with some successful changes at school my son has settled in much better and is no longer wanting to die or having intrusive thoughts.
I’m concerned that this process is going to take so long that he will be in year 10 or 11 before they even get to assessing him, so I am thinking it may be beneficial to have a private assessment.
My GP and CAMHS have told me that they do not recognise private diagnosis for autism.
Can anyone tell me what the benefits of an NHS assessment is over a private one? He’s never been on any medication or anything so I can’t see that would be an issue. He doesn’t have an EHCP or anything. Does an NHS assessment open up help at school that a private diagnosis wouldn’t? If he develops anxiety (for example) and needs medication surely he’d be treated on the NHS for that.
Finally, can anyone tell me, can I try for a private diagnosis whilst waiting for an NHS one? And where would I go for a reputable one. Before high school I had decided that a diagnosis wasn’t important but I’ve definitely changed my mind and feel it’s worth exploring to help my son understand himself in the future.
If you’ve got this far, thanks so much for reading. I’d really appreciate your advice. Thanks.