To get back to the original question. I see two possible ways for the Irish to conquer GB.
1) In ancient times (like the say in Star Trek, meaning before the Irish got totally behind in weapons tech and England developed into a strong nation, very roughly before 1000) the Irish get cocky and start raiding England as pirates. They are lucky and the natives are weakend by internal trouble and/or other invaders attacking at the same time. Beeing unexpectedly successful the Irish establish a permanent presence there and over time conquer the place.
This not so ASB as it sounds as it is basically what the Anglo-Saxons, Bretons and Normans did, the Danish almost did and the Irish did to Scotland.
The future is butterflies. Maybe not much changes (if only a small ruling class actually settles in GB), maybe all is undone by later invaders, maybe the UK becomes celtic.
A permanently unified state is unlikely because:
a) Ireland itself was never unified for a very long time
b) Never work for any of the other invaders
2) At some point during the medieval times (latest, allready unlikely chance: Civil War), after England had allready begun conquering Ireland. England / GB descends into a massive bloody battle for the crown. Through heavy luck, successful marriage strategy and alliances with english nobles an Anglo-Irish/Irish noble succeds.
But England can not permanently be ruled by the Irish (look at population and ressources). So either he is overthrown quickly or he adopts English customs and culture. So 100 years later everything is back to normal except for Irish symbols featuring more prominetly in the GB CoA and maybe the Irish a little better treated and motre loyal to the crown.