The fact that the father of the baby stood by her, as it were, would have gone down well. Any records which have to do with Institutions and anything to do with the health board are normally, as far as I know, kept by the Cork Archives Institute, which is in Christchurch, South Main St, Cork City- Tel 0214277809- but they would not look them up for you. If they have the records, and you will need to phone to find out, then you wil either have to look at them yourself or try to get a researcher to do so on your behalf. Being a Convent and a Catholic Institution, there is a possibility that the archive institute might not have the records, but you could contact the Diocese of Cork and Ross (they have a website) or, perhaps, the Good Shepherd Order itself, if it still runs a convent in Cork or elsewhere in Ireland.
Of course we are assuming that the Convent was indeed her place of residence, but it seems likely on the balance of probabilities that it was, especially if she was pregnant before marriage- a very big issue indeed in those days. The Convent was also an orphanage and the Cork playwright, Frank O'Connor has written a harrowing and very moving account of his mother's upbringing there in a short story entitled "Mother". If you can get this story, it will give you a great many details of life in the Convent and it paints a wonderul pen picture of the area around it, an area I know very well and love very deeply. It is a magical area, full of memories and stories for me, and I love to go there and walk its streets and lanes. The story would also give you a good idea of the kind of life the young child born to your family in that Convent likely escaped, owing to that father playing his part.
The Reuben immediately makes one think of Jewish roots, but it is also a name used in Protestant circles. When I first read the names of your ancestors, they struck my Cork eye as having much more of a Protestant than a Catholic look about them- but that is just instinct more than anything else. I may be wrong. Growing up in strongly Catholic Cork, where Protestants were very thin on the ground indeed, one was always immediately aware of Protestant sounding names! But that would not fit in with the possibility of your ancestor being in that Convent.
Ceallachain