First, not all service records will have survived unfortunately. Alfred (or whatever is actual name was) seems to have been one of the unlucky ones. I've often come across individuals shown on the census as 'Army Pensioner' but cannot find them in the online service records.
As to the marriage of Alfred Thomas Oyford - the actual surname is Byford and his father was a Printer. His wife was Alice Treloar (not Ireloar). However, it is a very poor copy and was somewhat difficult to read although the name Byford is quite clear. Her fathers occupation looks like Pension Officer to me - found her with him in 1881 and he's recorded as Navy Pensioner (surname transcribed as Preloar).
Your Alfred Thomas Offord was, as has been established, brought up by his maternal grandparents - he may never have seen his father again and would only know what his grandparents told him which was probably not much. Such as, 'you were named after him' so as he was Alfred Thomas then he believed that was his father name. As to 'Clerk', I see this time and time again when I look at service records - men often shown as 'Clerk' when they enlisted and again when they were discharged. I don't know if it had the same meaning then as it does nowadays.
Civil registration did not begin in Ireland until 1864 so that's why you wouldn't find a death entry for Eliza Offord there in the 1850's.
Sorry to be so negative but unfortunately things are not as cut and dried as we would like them to be.
The one positive now is that a) we know AT's mother was Eliza Neather and b)that she married 'Albert' Offord in 1850. Obtaining that marriage certificate has to be a necessity to ascertain the name of his father and to hopefully clarify his own christian name which is Alfred when his son is born and when said son marries, Albert when he marries Eliza Neather - not being able to see the original records online we have no idea what they actually read only what whoever transcribed them thinks they read. Clearly there is some confusion over his christian name which is why I think it could actually be 'Alban' as previously mentioned since a) Alban missing from UK census in 1851 and b) was a widower when he remarried in 1858 although no previous marriage in that name.
Annette
