Given the perimeters, an English heir's title could be anything. Yes, Ireland was thorny issue and I don't know of any medieval English king or king's son who actually succeeded in making good any claims on it. But, if such a thing ever were possible, I see no reason why a title for the heir is out of the question. There's no sovereignty issue like with Aquitaine, but theoretically the king could relinquish rule of Ireland to his son if he screwed up royally or required a puppet. So it has diplomatic value theoretically.
a surviving York dynasty would actually be useful in this regard. Since a) the Yorks were the primary royal landowners in Ireland (Edward IV was 10th Earl of Ulster, and had also inherited the claim to the earldom of Louth via his Mortimer ancestry) and b) most of the Tudor "repression" in the late 1490s/1500s was to put down the Irish who had backed Lambert Simnel (Warwick was Earl of Dublin and one of the "theories" about the dukedom of Clarence is that it was tied to Co. Clare rather than the honour of Clare in England) and Perkin Warbeck. Now, the wheels on the royal bus in Ireland started
coming off with Richard III's seizure of power in 1483 when the Burkes rebelled (AFAIK, this wasn't related to Dickon's coup, but a separate event). Unfortunately, the whole "Richard III-Bosworth-Henry VII" drama playing out in England meant that it wasn't dealt with quickly. If there hadn't been a "distraction" (say Edward IV survived), the then Lord-Lieutenant of Ireland would have had royal backing to deal with the Burkes (after all, once the Burkes seized the royal town of Galway and demanded ransom, the king would have little choice to get involved). Instead, there was the Burke's terrorizing pro-royalist Connaught, then the Tudor repression of the pro-Yorkist Simnel-Warbeck uprising (as seen by the fact that Poynings Law passed with the vote of only three
Irish peers- the remaining lords hadn't been summoned due to involvement/support for Simnel-Warbeck, and only one of those
is known to have been incapacitated due to his age (the Talbot earl of Shrewsbury who held the earldom of Waterford was also
not summoned back from France AIUI), culminating in the royalist victory at the Battle of Knockdoe in 1504.
So...just a thought: Edward IV survives. Hears of Burke's seizure of Galway. Sends trusty Richard of Gloucester to "deal with the Irish" (after all, lets give credit where credit is due, Dickon was
capable, having been sent up north to deal with the Scots). Then, we go Ricardian and Edward IV and his sons die while Richard is in Ireland, leaving Richard III as the last Yorkist male standing. His son, Edward of Middleham (who inherits dad's martial skill as well as grandpa Warwick's political skill), and he gets made "duke of Ireland" by his dad and left as Lord Lieutenant in Dublin. Later, Edward succeeds as "Edward V, King of England and
Ireland", and instead of sending heirs to Ludlow (a practice only started by Edward IV), they send the heirs to Dublin instead.
Least, that's my interpretation. Those more knowledgeable about Irish history of the time (Burke's, Simnel, Warbeck's rebellions can correct me).