With most auto policies, it will pay for any medical treatment due to an auto accident, under medical payment coverage or PIP.
The hospital or ambulance will (not) ask about your car insurance, only if you have health insurance, but either way will bill (you) the patient who sought care due to this accident. With most health insurance, if you have auto insurance, won't pay, (unless) your auto insurance denies it.
Until the auto insurance gets copies of all of the bills, then they can pay. The problem you have is that you (assumed) the auto insurance will pay after that 1 phone call, but they cant (until) they get copies of the bills from you.