I'm quite convinced that, because of its widespread popularity and style, it was issued on sheet music some time between 1870 and 1900.
It is nearly always linked to The Royal Horse Artillery and was very likely written to be sung by them. Bert may well have been right and Ed Harrigan perhaps wrote it. I can't really think of a reason why he would make this up.

Off the top of my head Harrigan and Hart were a big hit in New York having come over from Britain. He may have written it before he went to New York.