Emergency situations, what to do?

This is to plan in case an emergency happens.

Let's say hypothetically something happens to one of my cats while I'm away on vacation or out somewhere.

A fire could be one scenario, as I have a sticker saying about the animals in the house. It says to list an emergency vet contact on it, but if I'm out and can't be reached does that mean they bring the cat to wherever is listed? What if something happens after hours, and the vet listed is closed, what are they supposed to do?

Other scenario is away on vacation and someone is watching the house and feeding the cats. If something happens I don't think the people would bring my cat to a vet in an emergency, if that's the case but something happens and the cat needs care, but your out of state, do vets come to the house to pick up the cat since you are unable to if you call and explain the situation?

I never thought about this stuff until now and I realized I'm not sure what I would do if there is an emergency!

Our cat sitter is extremely reliable.  We have given her the contact info for the vet and for our next door neighbor.  We watch each other's places when either of us is out of town.  Our neighbor also has our cat sitter's phone number.

In case of an emergency, our sitter or our neighbor would take the cat to the vet.  

The sitter would be my elderly grandparent who came every other day to the house to add more food in feeder and clean the litterbox. However she lives 45 minutes away and doesn't stay at the house, and none of my neighbors talk to us or even have our phone number.

Our sitter knows which vet we take Mogli to and would take him there at the slightest sign of illness or injury. She also has our in-laws' and a neighbor's phone numbers in case she herself has a family emergency and can't come feed him. Both have keys. The vet has already agreed to bill us rather than demand money up front if Mogli needs care while we're away.

Around here, if the fire company or police rescue a pet from a burning building they take it to an ER vet if there's any indication of injuries, to the shelter if there aren't. The police have a key to an emergency indoor kennel at the shelter and will put the pet in there if the shelter is closed, along with details of what happened. They generally also call the shelter head, even if it's 2 a.m.. Whoever opens up in the morning checks the emergency kennel before entering the building.

My vet has my credit card on file so my cat sitter doesn't have to worry about paying and my sitter has instructions about taking my cat to the vet if need be, and contacting me prior if possible.