In case of a medical emergency such as a stroke, heart attack, etc, and you are wearing a medical alert bracelets then the initial things regarding the kids can get complicated. Remember that medical emergencies are often segregated from trauma emergencies.
Anyway, considering a medical emergency, if the EMS crew arrives first, the EMTs can start treating you, but transportation normally gets delayed until there is someone to release the children to. Even in a cardiac arrest, the EMS crew can not leave the children unattended. If you are conscious, and there is a neighbor nearby that you consent for the children to be released to, then that is one option.
Usually, the EMTs can begin care without transport until a police officer, sheriff's deputy, or other acceptable responsible person arrives at the scene to take responsibility for the children such as an EMS supervisor.
Ultimately, an ambulance, police car, fire truck, and hospital are not appropriate places to house children. In a perfect world, the next of kin would be contacted who would agree to take responsibility for the children. If that is not available, the police will get involved and Child Services will take responsibility for the child. Depending on the jurisdiction, Child Services will take the child/children from the scene, or from the police station as quickly as is possible. Medical bracelets can be a life-saving device as these are some of the most common yet major medical conditions. Every third person is suffering from these diseases.