I can't see a baptism to Henry and Mary -I can see one for a Henry and Elizabeth Halton.
Nuneaton 6 June 1818
William s of Henry and Elizabeth Halton Common Bricklayer
Rhoda Orton 6 Sep 1807 married 18.1.1831 Thomas Dagley and moved to Chapel End Hartshill
More on this, so I decided to focus on Elizabeth based on the above. I managed to find a burial for an 'Elizabeth Alton' in Mancetter records for August 23rd 1840. Abode was Hartshill, age was given as 59 (1781 estimate).
Pretty sure that this was the one, I ordered the death cert - Elizabeth Alton, Female, 59 Years, Wife of Henry Alton Labourer, Rhoda Dagley present at the death, Hartshill. As Rhoda married Thomas Dagley in 1831.. This is likely the wife of your Henry b. 1777.
I've yet to find her maiden name but I'll keep looking - We have an estimated birth now, so hopefully this will help.
Thanks again to ciderdrinker for knocking part of the wall down.
EDIT: I can now see that ciderdrinker also identified the 1840 burial