They would still have some way to go. Catholics are still a minority and more likely to view themselves as British than protestants likely to view themselves as Irish. You also have to remember that many devout Catholics are horrified at the secularisation in the republic, and I have heard of some formerly staunch Republicans stating they will vote DUP because their Christian values are more in sync than with the increasingly godless republic. The northern Irish are still very religious compared to the rest of the uk and Ireland.
the republic will become islamic soon enough