Brian, good examples.

But what percentage of the tunes in, for example, Ford's "Traditional Music in America" 1940) are actually British or Irish? Certainly not "most"?

Here's a list of tunes played at the Old Fiddlers' Contest in Dallas, as reported by the Dallas Morning News, April 14, 1901:

‘Tom and Jerry,’
‘Village Quickstep,’
‘Granny Will Your Dogs [sic] Bite?,'
‘Natchez Under the Hill,'‘
"First Four Forward and Back,’*
’Brilliancy,'
‘The Gray Eagle,'
‘Jennie on the Railroad,’
‘Miss Sallie Gooden [sic],’
‘The Girl I Left Behind Me,'
'Old Red,' *
'Clay’s Favorite,’*
‘Durang’s Hornpipe,’
‘Miss McLeod’s Reel,' ‘
'Scott No. 2,’*
‘Drunken Hiccoughs,’
‘Arkansaw Traveler,’
‘Wagoner,’
'Slapjack and Lassengers.’*

Additional titles from the 1900 Dallas contest:

'Cotton-Eyed Joe'
'Money Musk'
'Forked Deer'
'Dixie'
'Culpepper'*
'Gilderoy'
'Possum Up a Gum Stump'*
'Black-Eyed Susan'
'The Devil's Dream'

Other sources from the 1890's and early 1900's mention

'Mississippi Sawyer'
'Old Joe Clark'
'Pretty Little Liza Jane'
'The Eighth of January'
"I'm Gwine Down to Town'
'See Catfish Going Up Stream'*
'Cindy'
'Bonaparte's Retreat'
'Sugar in the Gourd'
'Brown Jug'
'Downfall of Jerico'*
'The Ship that Never Returned'
'Listen to the Mockingbird'
'Haste to the Wedding'
'A Hot Time on the Old Town Tonight'
'Irish Washerwoman'
'I Love Somebody'
'Annie Laurie'
'Old Zip Coon'
'Lannigan's Ball'
'Leather Breeches'
'Hell on the Wabash'
'Hell Broke Loose in Georgia'
'Old Rackensack'*
'Rack Back, Davy'*
'One-Eyed Riley'
'Fisher's Hornpipe'
'Nine Mile Island'*
'Bonnie Kate'
'Soap Suds over the Fence'*

Not a complete list, of course; just those I've noted over the years as I've encountered them.

Of 58 titles, I recognize about 46. Of these, 14 are clearly from the UK.

For convenience, I've starred the 12 unfamiliar titles. Most sound characteristically American.

Anyway, from a presumably random sample, 14 of 46 tunes appear to be of British Isles origin. That's a little less than 1/3.

More than I expected, but far from a majority. If the unfamiliar American-sounding titles do belong to American tunes, the proportion falls to roughly 1/4.

Julia, I'm not a fiddler. Just a skeptical pedant who doesn't like taking things for granted.