You have lost me again. To help you we need to get the facts and sequence straight. Is this what happened
1. A birth was registered in the name of Josephine McCann, child of Florence McCann. She was born 1 May 1934. And this certificate was issued in 1934. Is that correct and do you have the actual cert or have you tried to get it..
2. A second cert was issued on 21 Jan1966, in which the name of the child was Josephine McCabe and the mother to Josephine McCabe.
GROs do not gratuitously change names unless someone pushes them. A key question here is why would they have reissued the cert on that date. You mother was 31 and must have asked for the change. She would have had to provide proof. GRO will not make a change just because she said there was a mistake. What proof would she have supplied. Your mother must have had some "paperwork", what happened to it. Alternatively you could try to get from GRO reasons why the change was made and what paperwork they hold.
3. If the Reeves are stating the truth, and they registered the birth (the cert should say who registered the birth), the implication is that they did not know your grandmother very well. We could research the Reeves and see if there was a connection, but this is looking at there being no family connection.
4. Did you grandmother, "Josephine McCabe senior" then just disappear without trace.
5. "3 patrons church hold my mother's baptism cert and st. Kevin's hold her confirmation. " Do you have those, and what name do they have
6."the GRO still has not updated my mother's birth cert" The fact that a new cert appears to have been issued in 1966 to "Josephine McCabe". You need to check the 1966 GRO entries to see. The online records we have access to only go to 1958. You are in to the land of 2 "valid" birth certs, GROs will only release one of them, otherwise a person could legally have two passports in 2 names (reason UK GRO gave me). The original birth in 1934, registered as McCann will always remain, but they will refuse to issue a cert unless you supply a death cert to prove that you could not use both birth certs.
7. Reading what I have written, it appears to me that you have little "proof" that your grandmother was actually called "Josephine McCabe".The Reeves registered it as McCann. The baptism was presumably done by the the Reeves as well. So where and from whom did the new info come in 1966 to make the GRO change the name is the key.