Which doctor do you take a child's sleep problem to?

Start with the paediatrician: the general picture first, the referral second.

Then it branches. A neurologist if there's reason to suspect a neurological cause. An ENT for snoring, mouth breathing, pauses in breathing. A gastroenterologist if reflux is suspected. A sleep consultant is for the other case - health is fine, and what's broken is the routine and the habits around falling asleep.

And a short list that skips the queue entirely: pauses in breathing, seizures. Those go to a doctor now.