Thank you for those details. We can see more clearly now what happened
1. Regarding the birth, it does seem that the mother signed into the Rotunda as "Florence McCann".
The hospital registered the birth as such, and there was presumably no need for a Birth Cert for many years, so the Reeves never spotted the mistake. They only did so in 1958 when your mother needed a Birth Cert
No "error" was actually made, the Rotunda were just using the data the mother gave them
2. There was a baptism on 23 May 1934. Had Josephine been "given" to the Reeves by this time. In other words was the baptism done by the natural mother, or had she left the scene by then.
Also the sponsor is a clue. Is she known to the Reeves or to the birth mother, or both.
I did wonder if she was the birth mother, but the fact that the Reeves later had the change from McCann to McCabe militates against that. They just would not have bothered if the real mother had been Bridid Molloy
3. The Reeves seem to have thought about registering the birth (again) in 1956. You say you have some papers. What happened here. Were the papers drawn up but never submitted. Or submitted and rejected.
4. The Reeves did know who the real mother was and had no doubts that it was McCabe, and felt strongly enough to get the original registration changed.
It perhaps indicates that they were friends/relations rather than just neighbours.
5. So when it comes down to it the only real clue there is to work on appears to be what is the relationship been the Reeves and the birth mother.
Did the Reeves have any other children?
6. Sometimes in these stories, the birth mother flits in and out as an "aunt" or other relative that your mother might have known. Anything come to mind here that might be worth mentioning?
7. Given the birth mother signed herself in to the Rotunda as "Florence McCann" of "115 Hollyfield Building" you need to look at who was living at that address at that time. Was there an actual Florence Mccann at that address, or was that "Josephine McCabe's" address, or is it fictitious?