WAV, I doubt you'll hear "My Bonnie" at a folk club, just as you're unlikely to hear "Danny Boy" or even "Wild Rover" - they're too well-known and overdone to be attractive to most folk audiences, who consider themselves too sophisticated for such material.

Shanties are literally international - they came from a working community made up of all races and nationalities. The ones we are familiar with are those sung in English, but there are many others in other languages. The musical and textual influences are likewise extremely wideranging, as you would expect from a community which was not only multinational in composition but extremely widely travelled.

Apart from being in the English language, they have no particular connection with England any more than they have with anywhere else. They are not part of an "English" cultural tradition. However they are part of a maritime tradition which England (and the other countries of the British Isles) had strong links with, and most singers have no difficulty including them in their repertoire.