If the Angevin-Normans and later the English had never tried to annex Ireland at any time between he 12th century and the French Revolution, how do you think Ireland would have developed?
Ireland was going to deal with Britain sooner or later.
Even during the VIIth century, it had its fair deal of interaction with Anglo-Saxons (up to one campaign in Ireland for royal succession matters).
With the Viking Age, these pre-existing connections only became stronger (Hiberno-Norse settlements being largely tied with Jorvik, for exemple).
With all the regional proximity, already established links, etc. you'd have to make Ireland fantastically isolated to shield it against insular intervention.
Butterflying away Noman conquest of England (which is not too difficult) could delay English interest on Irish Sea, but not that much long : Late Anglo-Saxons were pretty much railed against Cymric principalties already and an pretty obvious nest of mercenaries avaible to settle Irish political disputes (IOTL, Anglo-Normans were called by Irish kings against their opponents. Which was retrospectivly a bad idea* but a pretty obvious one).
* While not exactly new : it had a long and prestigious history coming from Celts and Caesar to Byzantine and Latins, passing by Visigoths and Arabo-Berbers
As soon as an english noble would settle and dominate at least part of Ireland, or if an Irish king ask or became tributary of England (which could happen easily, see the Scottish exemple in Early MA), you'd have room for English intervention.
Your best chance is to see a lasting unification of Ireland (and that's not going to be easy, would it be because it would be in no-one interest) while England is busy elsewhere, and manage to fend off most of smaller expeditions.
Something equivalent to Davidian Revolution seems necessary : English nobility and elites would have far less room claiming of a semi-Christian anarchy to intervene with ecclesiastical blessing.
Eventually, it could look a lot as Scotland : definitely not shielded from wars with England, and far from being immune to its political influence, but somehow unified and distinct.
Toirdelbach Ua Briain, king of Munster and one strong claimant to High Kingship (or his son, Toirdhealbhach Ua Conchobhair) may have been a good candidate for...let's call it
Turloughian Revolution with ecclesiastical support, before Anglo-Normans lords could take control of Leinster.
I'd think it would pass trough, as Scotland, a period of de facto vassalisation of Ireland by England, but that's the closest I could think of.
Eventually, Ireland could manage to tie direct relations with the wider Latin Christiendom, bypassing partially English influence.
Would it have united by 1800, or would it still be a patchwork of independent states?
What would it be like in 2015?
Given the butterflies, it's near impossible to say without much wishful thinking.