The Mainly Norfolk page for this song does an exceptional job chronicling the recording history of this song and its closely-related variants. The earliest print versions date to the first decade of the 20th century. Rod Stradling commented on Sam Larner's 2014 anthology Cruising Round Yarmouth:
I can’t think of a song, supposedly about a real event, which goes by so many titles—The French Privateer, The London Man o’ War, The Irish Captain, Lord Exmouth, Warlike Seamen, Summerswell… the list goes on and on. Moreover, the date given in the song’s usual first line “On the 14th day of June, my boys, in Liverpool where we lay” seems to be different in practically every version!
So maybe this song isn’t about a real event—any number of Royal Navy ships were named Dolphin ; one laid down in 1751 was reportedly the ninth of that name. Whether this song is actually based on the exploits of a particular Dolphin is unclear. Nonetheless, it was pretty popular, with 67 Roud entries, and found in most southern counties of England, with a scattering in Scotland, Ireland and the USA. Yet despite 19 sound recordings, only that by the Copper family [on Come Write Me Down] remains available on CD.