From Michael in a positive mood.
Isobel found them on C1901 ( 4833-5-pages 2 & 3)at 8 Sunnyside, Bedlington ( this lay between the extreme "west end" of Bedlington and Netherton Colliery)
A look along the street to number 3 Sunnyside (4833-5-1) shows a Thomas H Graham, 33 born Earsdon, with wife Isabella born Lowick.
Both these Graham couples have a son (aged 2 yrs) called Anthony.
The English Naming Pattern, a convention that laid down rules for couples to name their children, was followed by many folk, although it was dying out by 1900s.
The first rule was that couples would call their first son after after the child's father's father. *
So it looked possible that Robert and Thomas were brothers and their father was Anthony Graham.
So going backwards to C1891, we find 16 yr old Robert and 23 yr old Thomas, both born Earsdon, living in Lowick, Bowsden Moor, (4270-116-9)
with siblings and parents Anthony and Ann Graham.
On C1881, in Alnwick, (5123-97-4) Robert W is 6 and Thomas H is 13, again with siblings and parents Antony and Ann.
On C1871 ( 5129-33-12) in Bates Cottages**, Holywell, Earsdon, Robert is not yet born, but Thomas is 3 yrs, with Anthony and Ann.
As to the registration of the birth of Robert William, there are a few candidates in the 1873-76 era, two of them Robert William, others just Robert.
Robert William Graham reg June Quarter 1875 @ Hexham DRO.
But the family never seemed to have lived in Hexham area.
Dec/74 Morpeth and Dec/75 Newcastle for plain Robert Graham. But again the family were not evident in areas covered by Morpeth or Newcastle DROs.
Leaving what I think is the best candidate for your Robert William Graham.......
Birth Reg, Sept qtr 1874, Tynemouth DRO.
The BMDs happening in Earsdon ( the village and the area covered by the Parish of Earsdon) were recorded within the Tynemouth DRO .
So if Robert William Graham was born in Holywell (see C1871) then his birth registration would have been made within Tynemouth RO ( but because it covered a wide area, there were eight sub-district offices, to save folk's shoe leather, including one in Earsdon Village.
Some Census Enumerators had a habit of recording "Parish of Birth", rather than "Place of Birth" ( My g/f was born in New Hartley Colliery village, but enumerators recorded his birth as at Earsdon.
* The second rule of the English naming Pattern was that a couple's second son should be named after his mother's mother. Rule 6 was that a couple's first dtr should get the name of her mother's mother
**Bates Cottages.... although this sounds like just a street or cluster of cottages, it was a small community in it's own right within Holywell ( Holywell was one of the eight "townships of the "ancient" parish of Earsdon)
Michael Dixon