I’m a foster carer. It’s not that unusual. If there were no one on hand at all when the paranedics came to take you then the police or social services would be called (ideally SS but if no time and you needed moving instantly the police would be involved and would care for your child until Social Services arrived.
m they would take steps to contact next of kin- your child’s father, any other contacts your child could give them. But if they couldn’t reach anyone your child would come to me. Ideally with a bag of bits from home, but not always.
I would look after them until family could be contacted or you were discharged. Sometimes that’s just a few hours - for eg a single parent admitted at midnight conscious in the morning and able to give contact details for the other parent. Sometimes that’s a few weeks - but if it were a longer hospital admission SS would definitely be looking for alternative family members to do the care, your child would just stay with me until it was sorted.
I would make sure your child was clean, safe, fed, comforted as much as they could be. If school age, I would take them to their usual school in the morning, and staff there would be involved along with SS in trying to track down a safe relative or friend. I’d be asking the teacher about any particular things I’d need to know about the child, assuming the only info I had was name address and school placement.
It’s really not an uncommon scenario. But once it had happened once, SS would encourage you strongly to find a decent back up option yourself - they should be able to help with that if you had a condition where it was a recurrent issue. But if you have a partner working away they would expect your partner to down tools and be the parent. Appreciate that’s not always possible - military service overseas for eg. But they would certainly expect them to be on the next plane home for any other working type thing.