Autism is a neurodevelopmental condition. In my country, from my own experience, it’s mostly psychiatrists and only a few neurologists who manage it. (Diagnosis, treatment etc) If you want a diagnosis you’re likely to be sent to a psychiatrist.
Most psychiatry residencies (where doctors learn about psychiatry and become psychiatrists) have a significant autism component, while that’s not the case for neurologists.
How does it work in your country?