What exactly happens after a child dies on the scene due to natural causes?

Depends. If a child is on hospice for known fatal issue, they are taken for burial. If it is unexpected there is almost always investigation and autopsy.
Depends. Various states have differing rules on the handling of death.If as you say this was an expected death, then the personnel handeling comfort care during the last stages should make the calls to the justice of the peace or other official & the person designated to pronounce & record the passing needs to handle such paperwork as needed.Your care providers should orient you to this process.