Abby, Jean Redpath says in her notes that she got the song from the Dransfields, so it's hardly surprising the versions should be so alike.

[1987:] Robin and Barry Dransfield [...] learned it from Johnny Handle who stumbled across it in a school song book. The "Merry Piper Song Book for Schools" [...] is the only printed source I have found and that publication claims that this is an "old Scottish Melody". (Notes 'Jean Redpath')