See also  Banbury Ale  at Greg Lindahl's  The Music of Thomas Ravenscroft

As MMario has just said, it was published in Thomas Ravenscroft's Pammelia (1609), but whether or not Banbury Ale turns up in other songs I don't know.  Banbury Cakes have been well-known for several centuries, and are probably still made.  Vaguely on the subject, the Opies (Oxford Dictionaty of Nursery Rhymes, 1951) commented:

"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."

Malcolm