"Billy Boy" is widespread in England, not only in the north east but in "London, Liverpool and South Wales" according to Hugill. The versions stem from "similar shore songs, of which there are many."

It seems to have been used as a capstan chantey (Terry), at the windlass (Runciman), and as a question and answer song (Hugill).
All the above from Hugill, Shanties from the Seven Seas.

I doubt and direct connection to "Caroline" since this is a common theme that is not confined to any one locality, and could have developed independently. "Billy Boy" was a common song everywhere English was spoken (still in school songbooks), and undoubtedly was known in English-governed parts of the Caribbean, but there is no evidence that it is related to the Caribbean "Caroline." Thread 32412, with lyrics to the 1927 song by Houdini.
Caroline, Calypso, Houdini