Well the only Stuart they have is the King they have already got- Charles I, and the Confederation of Kilkenny largely stayed loyal to him OTL so no big change.
Put the Duke of Lorraine (or son or brother of same) on the throne and they probably manage to inadvertently spark off a European war. Which may have butterflies for the English Civil War also.
Proclaim O'Neill King and there would be no massive change to OTL - he would have to flee to Europe to escape the Cromwellian conquest seven or eight years down the road. However, assuming he was a fairly decent King -and he was a capable military commander- there would be a strong Royalist streak in Irish nationalism -a "King over the water" even before the expulsion of James III and possibly a butterflying away of Republican nationalism in the late 1780s among the Catholic population after the extinction of the House of Stuart. Attempts to restore the House of O'Neill to the throne of Ireland. And, if the continental O'Neill line were to die out the O'Conor and O'Brien would have claims of their own.