I have found a tree on Ancestry, which I am now trying to verify, that may have my missing Thomas Bishop.
The tree has a John Bishop born abt 1790 who married an Ann Mulles (no dates given). They had a James and Thomas baptised in Fulbrook, Oxon in 1812 and 1813 respectively, then Mary Ann, Eliza and William baptised Wantage in 1815, 1818, 1822 respectively. Mary Ann married in Wantage in 1834 to Henry Rose and James married in Wantage to Lydia Hulcup in 1843. She was a widow, maiden name Farmer. Eliza seems to have several illegitimate children but did not marry. There is no further information on the tree for William or Thomas. But it does place a Thomas Bishop of the right sort of age in Wantage as it looks as if his family moved there between 1813 and 1815.
I didn't just find this tree at random, I have a DNA match to the tree owner, but through a different line (Hawkins). I have checked the Hawkins side on the tree and it matches with what I have also found, so it gives me some confidence in the research.
I have found a John Bishop of right sort of age in Wantage in 1851, born Northmoor, with William (but b 1836) John says he's married but no wife with him.
There is a John Bishop age 40 in 1841 with William and Eliza (but they seem too young). Looking at the image it looks like there was a 5 behind the 4, as if age 50 might have been scratched out.
Can anyone help fill in the gaps please.