Number 1863 in the Roud Folk Song Index. Beside the example in the Blann MS, Dr George Gardiner noted a fragment from a Mr Keely at Bath, Somerset, early in 1906: neither of those have their tunes. There is what may be a Nova Scotia example (unpublished) in the Helen Creighton collection.

The song appeared on broadsides; one can be seen at Bodleian Library Broadside Ballads:

Our Jack's come home to-day

Nutty has already pointed out a set at Levy; there are a further two there (search for "our jack"). Written and composed by W J Devers, and -judging by such date information can be found via resources like COPAC- apparently dating from the 1880s. It was evidently widely published (the University of Melbourne has sheet music printed locally) and presumably popular, so the early date of one of the parodies quoted above is not inconsistent.

See also the related thread Any more verses "Our Jack's Come Home Today"