On the other hand, the Normans in England certainly were Anglicized by the time of say Chaucer. They still spoke a lot of French but I think they definitely were tipping to an English identity. The Normans in Normandy were Gallicized before that, and I am guessing that the same thing happened in Sicily.
However if you bring in a bunch more settlers from Scandinavia, England and who knows, Normandy itself (or Gascony or Aquitaine) I think the more important question than what broad culture and language do they identify with, than whether or not they can create a strong unified rule in Ireland. Wouldn't be a more salient AH question whether that is possible or not?
It might not be possible because Ireland is just too poor to exploit effectively, grasping enough surplus to create a strong unified rule over the whole island; it could be it was left decentralized for Elizabeth and Cromwell to try to figure out how to rule it by sheer brute force because the poor produce favored hardscrabble little tenacious local clans.
If the basic problem of creating a strong central rule is solvable in the Middle Ages at all, more oppressors coming in might be key--or they might simply swamp the efforts of fewer and more politic Normans.
If it is possible, we could have a scenario where the subduing of the peripheral regions and hanging on to all the bits required a lot of English and perhaps more widespread Norman support, and the regime is now beholden to England and effectively subordinated to the English crown.
Or--in the course of making a fairly strong state, as best they can on skimpy Irish resources, they may become highly independent and declare a High Kingdom of Ireland (the "High" being a nod to old Gaelic tradition and a claim that the Norman monarchy in fact inherits the ancient High Kingship) and manage to defend it well enough to stay independent. Perhaps the Irish, even united, cannot resist full on British attempts at conquest, but perhaps diplomacy in various forms deters it--either the Irish ally with continental powers who pose enough of a threat that the English dare not antagonize them for the dubious gain Ireland would be versus the cost, or perhaps the Irish ally with England and deter conquest that way.
Either way, if the nobility were to be Gaelicized, what of it? They could not be so in the political sense or the damn kingdom falls apart, but they can in other respects and remain something different than they were OTL even so.
And if Ireland, perhaps pursuing the course of alliance with England, does get incorporated into a United Kingdom of some kind, the Normano-Gaelic aristocracy can become as Irish as they like, but Ireland's fate is quite different.
At least unless and until the English go Protestant and the Irish refuse to, which seems moderately likely to me. OTL the core of English Protestantism was the better-off parts and the poor country regions remained Catholic much longer and to an extent, indefinitely. Ireland would be that writ large.
Failure to be on the winning side in a religious civil war could bring troubles and woes on Ireland as terrible as OTL--in fact that's pretty much what did happen. But perhaps an Ireland with an old tradition of independent and unified rule can resist well enough to bring in allies from the Continent and thus prevail on their own island, or most of it, in time?