Will the ER do an emergency DNA test to establish familial relationship?
What will happen is that the ER will contact social services (in house and whatever county the child is in-- this might be known as department of family services or DFS) and make arrangements for the test to be performed. Turn around time for a private lab (maybe the family volunteers to pay for the test) may be 1-2 days. Something done through the state is going to take longer-- my brother quoted 7-10 days.
During that time, the child can be admitted into the hospital (but again, this will depend on how full the hospital is and may be unlikely if the child is not ill.) Or, the child will go into foster care until the test results become available and social services examines the home the child is going to. In my book, the child went into short term foster care-- this can always add conflict to your ms.