NHS vs. independent diagnosis - what's the difference?

Hi everyone - looking for a bit of advice here.

I have been on the NHS waiting list for an ASD assessment for about 11 months (Scotland). During this time, the opportunity arose to have a free independent assessment through my university. I agreed to that, and was subsequently diagnosed ASD in October. It was based on a 3 hour interview with a clinical psychologist, and I received a 6 page report detailing her opinion, plus some signposting, e.g. details of my local autism charity. 

As far as I know, I'm still sitting on the NHS waiting list. Do you think I still push for an NHS diagnosis even though I've now had an independent consultation? I'm curious if there are any differences between an independent and NHS diagnosis, e.g. in the assessment procedure, or type(s) of support offered. Also I'm not sure if my independent psychology report is valid proof should I ever want to register as disabled / claim anything, etc?

Lastly, I'd also like to know if an NHS assessment can identify a combination of conditions such as ASD, ADHD and dyspraxia. When I met the psychologist, she said there is a possibility I also have dyspraxia, but she wasn't able to make that diagnosis.

Does anyone have any thoughts or experience on this stuff? Any input would be much appreciated,  

Tintal