My two were diagnosed by the community paed.
Each area seems to have their own system but at ours you have an initial appointment with the paediatrician. It takes about an hour and ivolves answering lots of questions about your child while the Paed also observes them.
At the end the Paed may ask you if you have any particular ideas about what the problem could be. When I said autism the Paed said they agreed but that there would need to be a full assessment to make sure.
You then go on the next waiting list for a multi-disciplinary assessment. In the meantime the various people (child psych, SALT, OT etc) visit your child at pre-school/school/home to do their own observations.
When you reach the top of the waiting list you are given appointments to see each person - usually they will team up if possible and do each appointment as a pair. So, for example, the SALT and Child Psych might do their tests at the same appointment.
At the end there is a big meeting where everyone discusses their findings and the Paed will give a diagnosis if appropriate.
It didn't take all that long, certainly compared to other areas I've read about on MN. I think there was a wait of around 3 months for the first Paed appointment, and then a further 3 months to get the other assessments and dx.
The most recent was over 4 years ago though so I'm not sure if they still have the same system or not.