I think Prudie got the theology right. For Episcopalians and Catholics, once you are baptized, that's it. You aren't baptized into the specific religion, you are baptized to become a Christian. Converts to Catholicism who have been baptized in any Christian faith cannot be baptized again. Instead, they are confirmed into the Catholic Church. This is somewhat different from the way some other faiths view baptism.
But I'd see the grandmother's behavior as a huge, huge red flag that she won't respect other parental boundaries and I would deal cautiously with her, until she can prove that she will respect the parents' wishes about how they want their children to be brought up. And if she can't prove that, she'd never be left alone with the kids--not because of the threat of baptism, but because of what other boundaries she might cross.
As for emergency baptisms--in WWI, my grandfather performed a battlefield baptism. He and his buddy had been shot and were lying in the mud, waiting for rescue. His buddy was very badly injured. They had had many talks about Grandpa's Catholic faith. So Grandpa knew that his buddy had an interest in becoming Catholic. Grandpa asked him if he would want to be baptized and his buddy said yes. Grandpa used the last of the water in his canteen to baptize him. He died shortly afterwards.
That's an emergency baptism. When my nephew was born and the doctors thought he wouldn't survive more than 24 hours, one of the nurses in the delivery room, a friend of my SIL's who attended their church, baptized him right away. Then they were able to get a priest to come and baptize him a few hours later. Fortunately, Nephew defied the doctors' and survived. But again, that's an emergency baptism. Holding an already baptized baby under a faucet does nothing but get the baby wet.