With French Louisiana surviving longer, Cajun might have been a real, big mainstream thing.
I always wondered what European music would look like without the american cultural hegemony/influx of the XX Century.
There were some musicians/producers who claimed that they were trying to do this sort of thing. The British label ZTT, with music writer Paul Morley and producer Trevor Horn, actually had this as a quasi-agenda in the mid-1980s with their acts Frankie Goes To Hollywood, Propaganda and The Art Of Noise. The basic idea was: What if the European expressionist traditions - Dadaism, futurism, Cubism, etc. - would have influenced pop culture instead of American rock'n'roll? Another band that went back to some aspects of alternative subcultures of the Weimar Republic were German wave band
Malaria!. And there's of course early industrial bands like Throbbing Gristle, whose founder Genesis P-Orridge wanted to deliberately create a European, post-Industrialisation sound, as he said that the European experience was a different one than the Afro-American one which influenced the blues/jazz/R&B/rock'n'roll lineage.
The British Invasion (of the 1960s) was re-introduction of an American style to the USA.
Gospel, blues and jazz music all have their roots in African rhythms brought over by slaves. However, the blues were limited to "race" radio stations until the 1960s.
A few white American musicians like Jerry Lee Lewis and Elvis Presley where the firs tot gain air-time singing "negor songs."
Meanwhile, white, British rockers like Joe Cocker and Long John Baldry took up the blues, added their own licks and re-introduced the blues to the USA. In recent years, ERic Clapton et al have paid their respects to classic black blues musicians like BB King.
That'd make an interesting POD: if rock'n'roll wouldn't have developed in the first place, its "parents" country and R&B would take a different path. Musically, early blues and country records weren't a millions miles away, so it's not impossible that someone else would have taken up the Johnny Burnette/Elvis/Lewis-style formula.
Without American rock'n'roll and R&B, British popular music would have stayed in its trad jazz/swing and skiffle traditions, and, come the 1960s, Jamaican styles would have crossed over as well. So the quintessential British sound might not have been Mersey Beat, but a hybrid between folk, ska and skiffle.