The best practices regarding neurodevelopmental issues are inter and transdisciplinary, consider this example.
The pediatrician is usually the first practitioner to see the patient, he should suspect and make a screening test, a psychologist might then perform a diagnostic evaluation.
People with ASD have increased risk of epilepsy so an EEG might then be performed and interpreted by a neurologist.
If it's comorbid or otherwise a complex case, a psychiatrist might then come into action.
Many people with neurodevelopmental disorders will have some form of motor difficulties so a physiatrist might be called.
In short, there are many professionals that can and should be involved in the evaluation and management. There is no one "jack of all trades" practitioner for developmental disorders.