Is his birthplace ever given more specifically than Monmouthshire in the censuses?
Presumably you are aware that if he was born before the introduction of civil registration on 1 July 1837 there will not be a birth certificate.
1871 Census - Noted as Monmouthshire (Robert Aged 34)
1881 Census - Noted as London, Middlesex (Aged 45) - All the family have the same Birth Place incorrectly described. They were living in Battersea. Robert was Monmouthshire, Eliza Devon so imagine they just completed there current address whereabouts on the form.
1891 Census - Noted as Monmouthshire (Aged 54)
In the 1861 Census it only shows Eliza and William First son) on 1861 census still living Llanhilleth, Monmouthshire where the family went after marriage but Robert not with them. Only a hunch, but he’s possibly elsewhere working or searching a job as the family then move to Battersea where they are for the 1871 Census.
Aware of the 1837 cusp but live in hope that he may have been the 'right side' of the year OR the experience of those here could locate the Parish for a record search in the Parish Book.