ODNR refers to most of them, but should not be taken as necessarily endorsing any. Note that, of the 'Fiennes' lady story, they mention that 'the 19th Baron Saye and Sele' [Fiennes] '... suspected that his father, a noted wit (author of an autobiography Hear Saye) himself invented the Fiennes version.' The Opies suspected that the 'Coventry Cross' version was comparatively recent, and also point out that in Gammer Gurton's Garland (1784) and most other early examples, the lady is 'an old woman'.

Many of the quotes from (unidentified) websites in this discussion are (unacknowledged) quotations from ODNR, though in some cases altered to exaggerate their significance. The Opies don't refer, of course, to the inevitable 'pagan origins' stories, which are merely romantic fantasy.

Joe stated earlier (7 years ago; this is an old thread recently revived) that the Opies 'guess the song comes from the 15th century'; they don't actually say that. What they do say is 'Again, though it would seem unlikely that the rhyme originated very long after the cross was destroyed, there were, in fact, other, inferior crosses at Banbury, and the memory of the big cross always lingered.' Bear in mind that 'the turn of the 16th century' means the end of that century, not the beginning: the High Cross at Banbury was demolished in 1600, as has already been mentioned.

David Miller's story is new to me, though. Whether or not it can be considered a possibility as it stands would depend on whether or not Amy Banbury's grandfather was old enough to have been alive in the second half of the 18th century, when we know that the rhyme was current.

There are other problems, of course. It seems that none of the 18th century earls of Banbury was called Jonathan. The title apparently lapsed in 1813; the hereditary peerage Baron Banbury of Southam being a later creation (1924) and not related to the original, being named for the surname Banbury, not the place. Banbury Castle, also mentioned, was demolished in the aftermath of the Civil War.

On the whole I'd think that particular 'explanation' to be just one of those interesting pieces of folklore that arise in families and which turn out, if looked into a bit, to have very little foundation in fact.

For more on the rather vexed history of the earldom of Banbury, see (for example):

The House of Knollys

Celebrated Claimants from Perkin Warbeck to Arthur Orton