as far as I know, your baby is checked out by a hospital paed, and then all later checkups are done by the family GP. Paediatricians are considered specialists; if there's a problem, the GP can then give you a referral to a paediatrician, as would happen with any other specialist.

ETA: I didn't mention MCHNs (maternal and child health nurses) You visit them at their clinic for the first year to have your baby weighed and to be told off for feeding too much/not enough/being on the wrong spot on the length/weightchart.