A long time ago in Ireland or in England someone (can't remember who or where) told me that this tune was the theme tune, played by Jimmy Shand, for a regular BBC radio program, and was picked up by Irish musicians from the radio.
I made periodic efforts to discover the truth of this story, asking many far and wide, with no luck.
A couple of years ago I found what I took to be the Scottish parent of the tune while browsing this page: http://www.raretunes.org/performers/will-starr/ Go down to the clip of "Glasgow Highlanders".
With this information I asked some aging Scottish button box players (i.e. about my own vintage), one of whom replied as follows:
Decades-old mystery solved. So there you go. You can all call it the Glasgow Highlanders now"The Glasgow Highlanders" was the theme music for the BBC (Scottish Home Service) programme The McFlannels. Think "The Broons" on radio, or "The Archers" relocated to Govan.
I remember hearing it when I was about 5 years old. I'm sure it was on on a Saturday night, probably immediately before "Scottish Country Dancing".