Beside white and blue, versions are also known with green cockades (there are probably others). White as a cockade-colour associated with Jacobite affiliation seems to have been a relatively late development; Frank Purslow (The Wanton Seed, 1965, notes, 122) comments, in respect of a Blue Cockade variant noted in Dorset, 1906:

" '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."

The reference may very well not be to official regimental colours at all. At all events, you will have to look at other possibilities. The best-known White Cockade tune, of course, is older than the song that gave it its name. Whether or not the cockade in the putative "original" song was white, we don't, I think, know; though it is the most widespread form. Anne Gilchrist wondered whether "orange and blue" (in the different context of Green Grow the Laurels) might refer to an alignment of British and Irish protestant forces -this in relation to the Jacobite associations of the white-, but I've also seen it glossed (again in the context of that song) as the colours of a wedding dress. So far as cockades go, orange seems to have been the colour of Nassau, and black that of Hanover. I don't know who blue, or green, belonged to.