ASB if it's the Congress of Vienna. If the 1798 rebellion had succeeded due to support by the French troops who never got there in OTL, and a lot of things went badly for Great Britain, there might be a temporary Republic if there was an 1802-like peace. But then I doubt there would be with such a big thorn in Britain's side.
Only way to get one lasting for more than a couple of years is for Britain to be successfully invaded by France, which is difficult, though not impossible. I suppose if Napoleon had done it, OTOH, he might spin Ireland off as a separate kingdom under one of his less well liked marshals.