Ireland didn't recognize the puritan government in England and operated more or less as an independent country for several years saying they were loyal to the king, Charles I, and after his execution/murder, Charles II.
Part of the reason for the English Civil War was Charles I had a Catholic wife and was appointing high chuch Church of England bishops and trying to force the Scots Presperterians to use the Anglican Book of Common Prayer, which they Scots considered to be the mass in English. James II was a Catholic and Charles II is supposed to have sent away the Anglican bishops on his death bed and had a priest administer last rites.
The Irish were reasonably comfortable with the high church monarchs and the puritans were hostile to Catholicism. Cromwell was tolerant of Quakers, Baptists and so on, who were fellow dissenters from the C of E and extreme protestants. The puritans may also have seen Jews as having somewhat similar religous beliefs.
The worst fighting and repression in Ireland and loss of lands of the Irish aristocracy had to not with rebellions against England as such, but with Irish support of the legitimate British monarchs, who were high church or Catholic. Similar things happened after the Glorious Revolution when William III usurped the thrown from James II.
Cromwell considered opponents rebels and did not take prisoners. Priests were systematcally killed. Many noncombatants were often massacred or sold as slaves. Interesting that people think that slaves in what is now the US and Bermuda are all of African origin. The Irish slaves were majority women, often widows. The African slaves were mostly men, as they had more value for farm work. If the slave owners didn't keep the Irish women to themselves, they often "bred" them with the African men.