Snuffy is right - there is no evidence of the Marlborough parody until WW I ... probably a defensive parody by you Poms after the Aussies sang Waltzing Matilda late into the night!
Any real existence of a Marlborough period song of that description would require that an English song was remembered by a dozen different informants in Australia, two centuries after it was popular ... but it had neer been heard by all the British collectors/anthologists of the 18th and 19th centuries ... ?!?
Interestingly, the first folk song known to have been swapped in Australia (by the European invaders, anyway) was about Marlborough! When La Perouse's French expedition sailed into Botany Bay. just days after the First Fleet's arrival, the officers fraternised ... and the only drinking song they had in common was Marlbrouck s'en va t'en guerre. Not only did the English and French officers sing it - but it is claimed that the the Botany Bay Aboriginals, in their canoes, could be heard singing the tunes, days after!