Without Home Rule, they'll definitely rebel against Britain in World War I or whatever it's called. Also, if that conflict even happens in Cinco De Mayo I expect the alliances to be more or less the same as OTL but with some differences. The Confederacy would likely stay neutral since there is very little they would gain from the war and there is no eternal grudge between America and Britain for diplomatic and trade reasons.They are definitely not under Home Rule, which was such a nonstarter IOTL that Gladstone’s push for it broke the Liberals and ushered in the dominance of the modern Conservative party. Thanks to TTL’s Sydney Affair the appetite for Home Rule is virtually nonexistent outside of a few Liberals like Lord Ripon, which of course has huge butterfly effects for the Liberals and Tories.
As for Ireland itself... we shall see!