Frank Purslow (The Wanton Seed, 1968) had this to say on the subject:

"A widely travelled song, the colour of the recruit's cockade changing according to the area in which the song is found or the political opinion of a singer through whose hands (or mouth!) the song has passed.  During the 18th century wars, cockades of their national colours were worn by the soldiers engaged.  The white cockade became the distinctive emblem of the Jacobites and this is usually the colour mentioned in Northern counties, where the song may possibly have originated.  "The Orange and Blue" (which also appears in the song Green Grows the Laurel,) may refer to the army of William III, in which case the "blue" version of the song must be the earlier."