Trying to summarize a bit, so far....

1780s-90s:

General references to African and New World Black work-songs, from Mali, Grenada.

1800s:

General references to African-American work-songs and their style, from Martinique;
Rowing songs from Georgia, South Carolina, Guyana, Surinam;
Windlass songs, aboard vessels with sailors incl. from Northumberland and Holland.

1810s:

2 stevedore songs from Jamaica that resemble chanteys;
African-American rowing songs from Antigua, Virgin Islands;
Singing and fife-playing at the capstan on a British war ship.

1820s:

Rowing songs, from Georgia, Virginia, St. Thomas;
A version of "Cheerly Men" for topsail halyards on a brig near Quebec;
Fictional capstan shantying in the Arctic; capstan (?) song of British tars in London; chant for pulling known to an ex-British navy man. [I've also seen another reference to the phrase "British capstan song" from 1825.]

1830s:

African-American rowing songs in Florida, Georgia, North Carolina, South Carolina, Guyana, and "West Indies"
African-American firemen's songs on the Ohio and in general
Cotton stowing songs in Mobile Bay
"Ditties" at the capstan on an East India Company ship
A song at the pump windlass on a transatlantic voyage
Songs "for capstan and falls" and for catting anchor, on brigs off the coast of California
A capstan song on a ship off the coast of Arabia
The adoption of capstan songs and sailors' "ditties" by locals of Tahiti and the Society Islands

Summary:
Through the 1830s, references to African-American work-songs for rowing continued to appear. Since the 20s and continuing through the 30s, cotton-stowing had work-songs. The work-songs of the Black firemen/stokers in steamboats emerge in the 1830s. These all seem share some features, though it is difficult to say how similar they actually were; some songs crossed occupations.

At the same time, there is increasing reference to the songs/ditties for capstan. Though the practice was older, new songs were being added, some of which were probably popular songs of the time, for example "Grog Time of Day," which crossed cultures. By the end of the 30s, there were songs for the pump windlass that may have shared a form with what is ow thought of as a halyard chanty form. At least one instance of the jump from Black rowing songs to (perhaps from?) deepwater sailors' songs --and then to Pacific Islanders! -- is found at the end of this period in "Bottle O." "Round the Corn/er" is another song that seemed to float around between cultures and occupations; it is quite unclear to me whether it had any sort of definite form at all, or if it was a common catch-phrase.

I still wonder if, by the 1830s, the African-American style work-songs that would come to typify much of the "chanty" repertoire had yet any significant influence on the deepwater sailors.

I'd love to hear more opinions on this subject -- on whether one feels that the sort of chanty forms typified by "Roll the Cotton Down" and "Blow the Man Down" and "Whiskey Johnny" and "Hanging Johnny" etc etc had entered the scene by the 1820s or 1830s.