Hi Alan,
I can't help with the birth, nor can I find Elizabeth in the 1851 census but have found Robert.
1851 living with the Davy family, High St, Wooler as a servant
1861 he was living 2 Goulburns Close, Newcastle and now married to Dorothy who was born Hexham c1837. Robert was 25, a Cabman, b Wooler and they had a daughter, Esther, 1yr, born Newcastle.
RG9; Piece: 3824; Folio: 17; Page: 27
1871 living Percy Street, St Andrew, Newcastle. Robert was a Cab Driver, wife still Dorothy and children Hannah 10 and William 5 both born St Andrew.
RG10; Piece: 5087; Folio: 46; Page: 12
1881 living Carlisle St, All Saints, Newcastle; Robert, 44, Coachman, widower; William F Gray 12; Elizabeth H Gibbon, mother, widow, 67, born Wooler
RG11; Piece: 5062; Folio: 135
1891 Elizabeth Gibbon was in the Union Workhouse, Elswick, Newcastle.
I have to close now but that will give you some extra leads, I hope!
Rae