Why print mattered. This is long.

Tunes that we would still recognize as reels began to appear in Scotland and England a bit before 1700. They progressively took the place of jigs, outnumbering them in the Scottish repertoire by the late 18th century. There are very few documented Irish ones from before 1800, and with so many Scottish ones available, it made sense for an Irish dancing master to use those rather than try to reshape Irish jigs.

Example: "Mrs MacLeod of Raasay". This has a definite point of origin around 1780. Mrs MacLeod was a celebrity (Johnson and Boswell made a point of meeting her). But the tune wasn't exactly new. It's the jig "The Campbells are Coming" put into 2/2. And that jig tune is at least 100 years older - but not as a jig: its earliest known form is as the tune for a Gaelic satirical song from the Macdonalds of the Western Isles, "I was at a wedding in Inveraray" or "How can I be sad on my wedding day", slagging off the Campbells of Argyll (and used in a spectacular musical protest at the Treaty of Union in 1707, when the bellringer of Edinburgh played it on the carillon). The Irish version of the tune now played is a bit different from the Scottish reel of the 18th century - parts reversed, and they killed off Mrs MacLeod's husband - but the changes are nowhere near as deep as those that had already taken place in oral tradition in Scotland before Mrs MacLeod got it. (The Irish usually play it in G rather than the original A, but versions in G were printed in Scotland in the early 1800s: the only real Irish innovation was reversing the part order).

This kind of story is common to a lot of dance tunes - origin in a song with an utterly undanceable rhythm that gets regularized over decades or centuries. It's quite likely that the original song form of this one was known in Ireland too - but it didn't even evolve into a jig there, let alone into the reel. The reel only got to be as widely familiar as it did because the McLeods adopted it as a kind of anthem and tune publishers followed. And the Irish followed them, a lot later.

The tune was first printed by the Gows in 1788, but their book probably had little direct influence - it was posh and expensive, sponsored by wealthy amateurs, and you'd have had to be a fairly successful pro to want it: "Mrs McLeod" was copied into a lot of manuscripts by fiddlers with short arms. It may have appeared in single-sheet publications - the Gows did a lot of that after 1800, and Breathnach thinks there might have been similar sheets of Scottish tunes published in Dublin, but there's nothing I can identify for this one. (Dance sheets are very poorly indexed by libraries). As far as I can see the first mass market publication of the tune in a book was by Cameron in the 1840s, and his books were affordable by anyone who had a fiddle (as were Middleton's a bit later, and Kerr's in the g1880s). The first Irish book publication (as "Miss McLeod") was by Levey in 1858 - Cameron would have been the handiest source, but maybe not directly (or Levey would have got the name right).

America probably got that tune from both Irish and Scottish sources, but the Irish version got into print there first, in Ryan's Mammoth Collection in 1882.

BTW there is a description of Scottish reels getting into Irish tradition in Breathnach's "Folk Music and Dances of Ireland", but it's not much more than a list, with no hint of process, and while his description of the Irish use of the tunes seems pretty good given how brief he has to be, Scotland is "here be dragons" territory - he doesn't track anything back to its origins within Scotland, and is so uninterested as to throw in Oswald's bullshit about Rizzio as if it was historical data. The main significance of his account was probably that it exasperated Fleischmann into doing vastly better.