Steve, the first I heard about this was the Christopher Suckling interview on World at One on Thursday. (Nelson's mother's maiden name was Suckling too.) He said that Nelson heard a chant by his crew after the victory in the Battle of Cape St Vincent on 16 January 1780, during the American War of Independence. It's not stated whether he heard it as a work chant or in another setting, a celebration perhaps. He wrote it out and sent it to William Douglas, 4th Duke of Queensbury, who "domesticated" it for Nelson's household, added a bass line and an instrumental introduction, and changed it to a "more refined" soprano melody.

With grog and Nelson both together
We'll drink and conquer in all weather

Christopher Suckling sang this chorus which, according to the Guardian report, Douglas added too. The photographs of the hand-written manuscript on the Museum of London site show the chorus and 2nd and 3rd verses, but not the first - and presumably original - verse. Suckling does say that, in the 18th century, there would not have been chanteys, but maritime work chants would have brought rhythm to repetitive work.