It's called "a caul" in English. I, in fact, was born with a caul myself, according to my mother.
Apparently, there was once a superstition that it was good luck to be born with a caul. Furthermore, there was a superstition among sailors that anyone who possessed the caul from a baby could not drown. in the opening passage of a famous novel by Charles Dickens, "David Copperfield," the narrator says: "I was born with a caul, which was advertised for sale, in the newspapers, at the low price of fifteen guineas," hoping that a sailor would buy it!
The correct name for what you call "the bubble" is "the amnion" or "the amniotic sac." When it breaks during childbirth, in English it is said that "her waters broke." (For some reason, "waters" is plural.)