It would be interesting to know why Christine Hendry thinks this is a 16th century piece, and where the above text came from. As Martin has pointed out, the historical basis is known, and dated broadside editions are all of the 19th century. It was evidently popular, as scandals so often are; the Bodleian Library has 14 copies from various printers:
Sharp couldn't have published the set he got from George Say as it stood; not because of any particular impropriety in that particular version, but because it was incomplete and barely coherent. It consisted of 2 verses, rather confused, of Fanny Blair; plus one-and-a-half that had somehow wandered in from The Deserter. As for "censorship", you have to remember that this was nearly a century ago, and Sharp was publishing songs to be sung, and was obliged to modify certain things if they were to reach the general public. It was not until the Lady Chatterley trial that it became possible to publish frank material outside specialised, scholarly works or the "yellow" press.
A few sets were found in tradition; Ralph Vaughan Williams got one from Peter Verrall, but as usual neglected to note the words; George Butterworth got them later in 1909 (Michael Dawney, Ploughboy's Glory, London: EFDSS 1977, 43). George Gardiner found it in Hampshire in 1907 (Frank Purslow, The Foggy Dew, London: EFDSS , 29-30) and Gale Huntington (Songs the Whalemen Sang, Barre 1964, 229-31) prints a text from the log or journal of the ship Java, out of New Bedford, 1839, set to George Say's tune. It has also been found in America. Number 1393 in the Roud Folk Song Index.