"Bye and Bye" generally means in the future (Bum Bai for all the people from Hawaii out there).
By and by; i.e. in due course or at the appropriate moment. Here's another very old expression, going back at least to the time of Chaucer. Originally Bi and bi meant "in order; neatly spaced" and was known to refer to Time as well as Objects. Thus something occurring at an appropriate interval after something else could be regarded as having a neat and tidy relationship to it.
Double E has something to do with a train. How that could be a "waterfall over my back" I don't know.