More smoke and mirrors a lot of 'almost' but not 'quite' right, though records are often hit and miss on ages etc
other possible entries I could see in the register for St Joseph, RC, Birtley were:
Baptisms
entry 1233
Born 17th February 1859, baptised 20th February 1859, Patritus Coyne son of Thomae and Brigitta Coyne (maiden name MULHEARN). Godfather Dominicus Rogers, Godmother Brigitta Rogers
entry 1234
Born 17th February 1859, baptised 20th February 1859, Maria Coyne daughter of Thomae and Brigitta Coyne (maiden name MULHEARN). Godfather Thomas Malone Godmother Brigitta Conry.
Birth year matches for your Mary who was probably a twin, I can only assume that Patrick died as he doesn't seem to appear on later census records- though nothing in the civil registration index to match.
entry 1412
Born 8th March 1861, baptised 17th March 1861 Hannah Coyne daughter of Thomae and Brigitta Coyne (maiden name MULCADDEN). Godfather Jacobus Mealy Godmother Brigitta Vasey.
Though the On -- (female) on the census return has the age recorded as 2 and a half months and the census was taken on the night of 7th April so if its the same child then the age doesn't quite match
There was a burial entry in 1881
Died 20th December 1881 Thomae Coyne from Wreckenton, buried 22nd December 1881, St Joseph , Birtley - there is a corresponding death registration on the GRO site, Gateshead district Q4 1881, age 60
I ran out of time, so didn't find anything for Peter, but I should be back there late next month so will have a looksee. I can also check to see if there was a burial for little Patrick.
As the registrar says, it was compulsory to register births back then, but there is nothing coming up that even vaguely matches these baptisms. I don't know what, if any, checks were made to ensure people complied with the requirement. These days there are so many ways that registration could be chased up, maternity care, hospital records etc etc but way back then nothing like that was available to the ordinary folk.
On balance though, the matching parents names and abodes on the records in St Joseph's register does make them (in my opinion) fairly possible to be the children you can see on the census records. At least these give you something to think about and decide if you think they are 'yours' or not.
Sorry I wasn't able to come up with anything absolutely conclusive.
Boo
