This was recently mentioned by a parent.
A high proportion of doctors were educated in independent fee paying schools. These schools generally don't accept children with disabilities of SEN, even if they are academically high ability. It has been known for children with ASD and Asperger syndrome to pass entrance exams for these schools with flying colours, but then get rejected at interview or end up being 'kicked out' in the first term for being weird or not fitting into the environment. This result of this is that such doctors do not have any classmates with ASD in school.
The parent thinks this goes some way to explaining why it's rare to find doctors with any real knowledge of ASD.
I'm very familiar with the private education system - you'd be surprised at just how many academically able / disabled kids they take. In fact, one very high-level school I know of has its sixth form stuffed with aspies - they will all go on to Oxford or Cambridge and be brilliant students - but are socially inept.
You'd be surprised at just how many doctors & consultants are undiagnosed aspies too - but because they are successful and 'managing ok' they don't ever think about it. If you think about their work environment, it's all problem solving and data collection - and how often do you hear "He's brilliant, but has a terrible bedside manner...." Spot the pattern.
Unless you are talking about crummy little institutions - like religious schools - the fee paying independent schools round my way don't like taking any weird or 'problem' kids or those with SEN from primary school.