"Sir Sidney Smith's March" was known (and written down) in C19th, and I'd guess it was at least contemporaneous with Sir Sidney, i.e. early C19th or even very late C18th, when Sir Sidney was the nation's most popular hero.

"The Robbers' Retreat" is included in the Naval Song Book, edited by Inglis Gundry circa 1945. It is included (as are a number of songs in the book) "by permission of Rev G. Goodenough M.A."
Rev G. Goodenough was a Chaplain to the Fleet. I suspect, (perhaps someone could confirm,) that Rev Goodenough compiled "The Naval Song Book" published circa 1906, and that Inglis Gundry incorporated songs from the earlier book (including, maybe, Robbers' Retreat) into the 1945 book.