Brian, no I am not connected to the family (just one of the many on Rootschat that help others when they can).
If you have a good look at the tree you refer to on Ancestry it's completely 'up the pole'.
First, they have 'your' Robert William Stockton born per census ca.1852/3 as the one who's birth was reg'd Sept.qtr.1849 - a) clearly no checks have been done as that child was bp.26/8/1849 St.Stephen, Hull son of a William Stockton and wife Rosanna whose maiden name was Frankish. A further check would have shown that this child soon after birth as their death is recorded in the same quarter/year - indeed the child buried on 29/8/1849.
As amondg has discovered, for some reason there was a change of surname from Stockdale to Stockton hence not finding your Robert Williams birth before - he was reg'd as previously stated in Dec.qtr.1852 Hull as Stockdale not Stockton but not baptised under either surname! Why the change in surname between 1852 and 1861 from Stockdale to Stockton I have no idea.
Robert Stockdale full age Potter (that's what it looks like) son of Robert Stockdale, bricklayer married Ann Lideman full age, dau. of William Lideman, labourer - witnesses John Lideman and Mary Postlethwaite - on 25/11/1850 St Stephen, Kingston upon Hull.
Four months later on 1851 census no trace of Robert senior but as amondg has posted his wife Ann is shown as Stockdale living with her widowed mother Mary, Mary Postlethwaite 16 (presumably the witness to Ann's marriage) and her illegitimate son Charles Parkin 8 (not 2). However, as previously suspected and now confirmed Ann Lideman had never been married to a Parkin - son Charles' birth reg'd as Charles Parkin Lideman in Mar.qtr.1844 and how I don't know but he was baptised as Charles Parkin 15/4/1844 Holy Trinity, Hull son of Edward Parkin (shipwright) and Ann as if he was born from their lawful union (which he wasn't). I suspect he was actually born late 1843 as his age on census nearly always ends with '8'. (Said Charles Parkin marries an Annie Hall in 1868 and states his father as Edward Parkin, shipwright (deceased). I think Edward Parkin died in 1849 so presumably he and Ann were together until then (without benefit of clergy). The following year she marries (quite rightly) Robert Stockdale as spinster Ann Lideman.
Ann Lideman born 14/11/1823 and baptised as previously detailed by amondg, dau. of William and Mary. Her 'partner' Edward Parkin b.1821 was the son of a Barbara Parkin, widow, who married a Richard Lambert in 1831. In 1861 Census Charles Parkin is with his paternal grandmother Barbara Lambert and in 1871 - now married to Annie - with his mother Ann Stockton and half brother Robert.
Back to Robert Stockdale later Stockton. At his marriage to Ann Lideman his father is shown as Robert Stockdale, bricklayer and there is such a person living with wife Isabella in Hull in 1841 and 1851. Robert Stockdale, bricklayer, widower married Isabella Good, widow, 22/11/1824 Drypool, Yorkshire. They do not appear to have had any children together so son Robert Stockdale/Stockton was presumably a son from Robert's first marriage. However, I don't know who his first wife was - one would assume that Robert, father of your Robert William b.1852 was bc.1820 and although I've looked at possible Robert Stockdale baptisms with a father named Robert none of them have been a bricklayer (farmer, stone mason and tanner) so don't know where exactly he was born, when, or the name of his mother.
Can you confirm that it definitely states 63rd Regiment, soldier on RW's marriage certificate as occupation of his deceased father and not just soldier?
The details I gave previously were for a Robert Stockton who was in the 63rd Regiment of Foot service no. 1115 who on FindMyPast's British Army Worldwide Indexes 1841 in Moulmein (Burma?), and in 1851 Ashton under Lyne and Limerick (Ireland) as you'd mentioned this regiment.
However, should we now be looking for a Robert Stockdale in military records??
Annette
Added - always checkout Ancestry Public Trees - they are submitted just from members such as you or I and never accept without checking that what they say is true. They so often aren't!