The starting point has to be the marriage certificate of James Mitchell and Sarah Realf.
This reads:-
Marriage solemnized at the Parish Church, Oare (sic), August 7th, 1876
between
James Nelson Mitchell, 21, bachelor, labourer of Oare, son of Thomas Mitchell, fisherman
and
Sarah Ann Realf, 20, spinster of Oare, daughter of Thomas Realf, labourer.
One complication is that the birth registration for James Mitchell, son of Thomas Mitchell, former fisherman, August 4th 1855, simply calls him James, not James Nelson. He is however exactly the right age. Given the number of other people in the family with Nelson as their second name, he may have felt that he ought to have that family name and taken it for himself.