Agreed Giok. The problem is that ALL the people of Britain, including most of the English, have been primarily Celtic, racially speaking, since Caesar and considerably before--if indeed this idea has any meaning. But culturally, most of England and Lowland Scotland accepted (or were forced to accept) an overlay of Anglo-Saxon language and culture, to the point where the original Celtic language(s) of Scotland, both Pictish (If indeed it was Celtic) and British, are no more. The Gaelic speakers in Scotland were, as you say, immigrants from Ireland.

So what do you call English-speaking Lowland Scots? In many cases, they have roots in all four cultures, Pictish (which may or may not have been "Celtic,") British, Anglo-Saxon and (Irish) Gaelic. In some cases, of course, they have ancestry in France and elsewhere, which have similar histories of Celtic base and Germanic/Roman overlay.

The extent to which they are "Celtic" is thus debatable, but by no means can the claim be dismissed.

Funnily enough, most of the MUSICAL forms that make up "Celtic" music, Jigs, Reels, hornpipes and the like, originated elsewhere anyway. So even if the people ARE Celtic, one could argue that the music is not "Celtic," but simply the Anglo-Saxon, Scottish and Irish variations on West European folk music, as carried by Scotch-Irish settlers into America.

Or one can just throw up one's hands and say "Celtic Music."