Charley, The Royal Navy seemed almost phobic about shanty singing, I am sure it has been discussed on here before. There was a suggestion, I am sure, that is borne out by the contacts I had myself with crews of various RN vessels that it was regarded as something that, if not actually forbidden, was certainly frowned upon on Her Majestys ships.
As to what might be a suitable holystoning shanty, I would have thought a "stamp and go" capstan would be fine