What if England's attempts to convert Ireland to protestantism were a complete success, barring the obvious catholic minority such as exists in Scotland - how might these butterflies effect current day politics? I suspect no partition of Ireland but would it still be in the UK or would the Gaelic/Celt nationalist still be strong enough to call on a war on independance?
The Society of United Irishmen was formed by Protestants after all.
One might start by having Henry Frederick Stuart, eldest son of James I & VI, survive to adulthood and reign for thirty years or so. Henry was described as "an obdurate Protestant" with strong Calvinist leanings. One can easily imagine him devoting considerable effort to suppressing Catholicism in Ireland.
Henry was born in 1594, and James died in 1625, so Henry would succeed at age 31. If he reigned for 35 years, that would be enough time for such a project - at least to the extent of making Protestantism the majority religion.
Of course there are other major knock-ons from this - probably no English Civil War, to begin with. Further down the road, the Catholic Church in America would be much smaller without Irish Catholics. Irish clergy completely dominated the Church in America in the late 1800s. This led to the formation of the Polish National Catholic Church by Catholic Poles who couldn't stand being bossed around by Irish bishops and having only Irish priests.
There would be far fewer Catholics in Scotland - most of them are descended from Irish immigrants who fled the famine for the factories of Britain. (Note that Glasgow's "Catholic" football team is "Celtic" with a green shamrock emblem.)
Irish politics would be decisively affected. In the Kingdom of Ireland, the Parliament was rigged to have a Protestant majority. There would also be the question of the Church of Ireland. OTL, it was tied to the Church of England and Presbyterian "Dissenters" were also largely excluded from political power. Under Henry IX, the Churches of England and Ireland would be more Calvinist, and there would be fewer "dissenters" as well as fewer "Catholics". Thus the Parliament would be substantially more representative.
It's still likely that Ireland would feel oppressed by Britain and be rebellious. However, the Home Rule question would not be complicated by the religious question. Many Ulster Protestants aren't just anti-Catholic, they're fanatics on the issue. Ian Paisley believes that Jesuits are literally agents of the Devil. When Home Rule was proposed in the 1800s, Ulster Protestants were faced with rule by a Catholic-dominated government -
very Catholic-dominated. That's what made Home Rule intolerable to the point of threatened rebellion in 1914. If Ireland is Protestant, most likely Ulster is as Nationalist or even Republican as the rest.