Just trying to look at all possible alternatives, could Ann have been the daughter of an unmarried mother, who then married a Buckingham, and Ann took this name ? ie, did this happen then ? (It seems to me, from other research, that back then people tended to keep their birth surnames.)
We note that from Ann's 28 May 1870 Death Certificate says age 77, (which possibly means 77th year), so, if correct, she was born in 1792 or 3, or 1794 - which "matches" the range from the 1841 census. These details were provided by the daughter-in-Law, not the son [John] (although there was probably no family difficulties (?) because they had 16 children from 1857 [another John], regularly, ranging from every 11 months to start until 21 months in 1881, when she died, of puerperal fever.)