In SEA DRIFT, Hercules Robinson (1858) recalls his days in the British Navy and serving during the Battle of Trafalgar, 1805. I don't find where it says when he first went to see. However, it is that time (i.e. 1805 or earlier) that he refers to when giving his "Sally Brown" song.

When I went to sea first, the bellowing of officers in carrying on duty was awful, and a strong voice was a gift greatly prized. Every officer giving orders used a speaking trumpet, and the men were not half restrained in the article of noise. They were not allowed to do their work with such a song as Dickens commemorates—
" Oh Sally Brown, Sally Brown, oh !
She won't have a Yankee sailor, oh!
Cos she loves the nigger tailor, oh!!"—[Da Capo.]

But short of this refrain there was a great latitude as to exclamation and noise. My Captain, the first, Sir Henry Blackwood, had a wonderful organ, and might be heard a mile off.

Well, so they were not allowed to sing in the British Navy. It is not clear whether this song was of that time period (i.e. on non-Navy ships) or if it came in later -- in time for Dickens to commemorate it. Was it this song that Dickens commemorated, or another, perhaps as in HOUSEHOLD WORDS, 1855-56, which has an out-of-context reference to "Drunken Sailor"?

In any case, though this has the name "Sally Brown," and its lyrical theme, its form is not of the stanzaic sort, with "roll and go" chorus, that we know of later. It looks more like a "Cheerly Man" form, which would be consistent with the earlier time period and Navy context, if so. It's not Maryat's "Sally Brown" of 1837, but it does appear like the iffy "common sailor's chant" sung on stage by Wallack in the 1820s. I'm going to file it as "circa 1805-1820s."