When my Ds was diagnosed with ASD, we were seen by three professionals (a highly specialised speech and language expert, a neurodevelopmental pediatrician and a clinical psychologist) and the process took three hours. This was with the NHS. My understanding is that best practise is that the Ados should be conducted by at least two people.
This is not to say your Ds's diagnosis isn't correct though. Does the private doctor who saw him do NHS work too?
As Imitatingdory says, the use of the word 'mild' seems strange. In order to reach the threshold for a diagnosis, a person must present with significant difficulties. These difficulties aren't always cognitive though. I know of DC with ASD who are at grammar schools and some who are at top universities. Autism is a complex condition and every person with a diagnosis has their individual make up of social communication/barriers to learning/sensory difficulties.
In terms of schooling, how old is your Ds? What school year would they be joining? I know that for the competitive private schools in London there may be some sort of entrance test (even at 4 years old). They will also want to see how a Dc copes in a classroom setting away from the parent. Being brutal (and knowing children who've gone through this system) many of these private schools are a business - the 'business' being to get the Dc into the top selective secondary schools. I have known Dc who have been asked to leave.
However, there are other private schools that are nurturing and have a different ethos. As ChocolateHoneycomb has said, there are some good private schools out there. There are also good state schools.
You will need to contact all the schools directly to sound them out. You will quickly be able to tell how receptive there are to Dc who may have additional needs.
Is your Ds in any kind of setting at the moment? If so, what have they suggested or said?