I've honestly never heard of babies born to nuns in convents, or of priests and nuns having sex. It would be pretty astonishing if neither ever happened, but they're not part of the consciousness of Ireland, so I don't think they can have been common.
However, I did hear of a priest who brought up three babies that were left on his doorstep - all girls. He was a motherly man, and the parish and his superiors simply ignored the whole thing. He seems to have done a pretty creditable job. The three grew up and left, and he was all lonely in the presbytery by himself, then one after another they came back with their babies - marriages having broken up or whatever - and he delightedly tied on his frilly pink apron again and took to his grandmotherly duties, and they all continued to live in the parish house, again with no one paying a blind bit of attention.
He must have been a truly exceptional pastor for his parishioners to have valued him so much that they could pretend not to notice. Lovely guy, apparently.