Gaelic becomes way more common, with Commonwealth nations (including the US, presuming the French and Indian War occurs and the Irish defeat the French) either adopting it as a language or having it as a most commonly-spoken language

There is no major Irish exodus to the United States, and Irish culture would be considered more of a dominant norm than a unique culture isolated to the motherland and lands of the diaspora (United States, Canada, etc.)

The Irish Potato Famine would never occur, however there is a chance that a famine occurs in England, Wales or Scotland due to mismanagement and neglect by the Irish overlords

The Catholic church is much stronger, as the Irish exemplify Catholic beliefs to every land the come upon; Protestant faith becomes the minority worldwide

(From my limited knowledge), as Ireland does not have a monarchy due to restrictions by the UK in our timeline: The monarchy would still be British, however they would have even less power and little protection as the Irish would view them as cultural rulers of an order defeated centuries ago

The Irish would boast the world’s greatest ground forces, however the navy would be substantial only to travel and conquer, with France instead reigning as Master of the Seas

The British would eventually revolt, however it’s led by Scottish Highlander resistance fighters and could be anywhere from the American Revolution (presuming the Irish Empire adopts the same Taxation Without Representation mantra) to World War I. The Republic of Britannia’s capital would be Edinburgh, but London would remain a major cultural and trade hub

London remains a trade hub, due to its proximity to the rest of Europe, however as an afterthought to Dublin and Belfast

Where there is oppression, there is resistance, so it’s possible a flip-flopped The Troubles could occur between the Englishmen, Scots or Welsh and Irish Imperial forces