Cecil Sharp got this song in 1909 from Sister Emma of Clewer in Berkshire, and originally noted her text as fine coddlin apple tree; the spelling was normalised to codling on later publication. The DT text, GOING TO BANBURY is as quoted from Sharp's MS in James Reeves' The Idiom of the People (1958), though the title has been abbreviated from that used by Sharp (and presumably Sister Emma), As I Was Going to Banbury. line is one of several small typos in the file.
In The Oxford Dictionary of Nursery Rhymes (1952), Iona and Peter Opie comment, "The fact that so many nursery pieces mention Banbury may, in part, be due to the energy of the printer, Rusher. Working at Banbury he often altered the wording to suit local patronage, but his influence was more than local. His juvenile publications are among the commonest chapbooks surviving today."
Sister Emma was the source of interesting versions of a good few songs, including LONG LANKIN, though regrettably the DT file, copied from The Penguin Book of English Folk Songs (1959) does not credit her.