First let me say, however, that while it is reasonable to compare it to the "Santiana" series, which invariably contains the phrase "on the plains of Mexico," even if the two songs are related, this "Bay of Mexico" is appreciably distinct.
The first document of this song I have seen is in a short story by Herbert Lawrence Stone, “The Reckoning: A Story of the Sea," published 1903. Though a work of fiction, the author seems to have had some knowledge of existing chanties and makes reference to them. This song appears to be referenced, by title only, in the following passage.
Soon the click of the iron pawl dropping into place drifts aft, then the words of "Down the Bay of Mexico" rise in loud, crude tones, followed by "Walk Her Round" and "West Australia," to the rhythm of which the shuffling feet keep time. The iron cable comes slowly in, a link at a time, grating harshly on the hawsepipe, the mate now leaning out on the bumpkin to watch it, now admonishing the men to "walk her round briskly."
For the other pre-Hugill documentation of this song, I am going to excerpt the original notes posted with my video.
An interesting trajectory related to this chantey begins, I think, with the 1935 field-recording by Alan Lomax of singers in the Bahamas. It seems that this set of recordings, which includes "Round the Bay of Mexico" as well as a version of the soon to be well-known "Sloop John B.", was a significant source for performers like the Kingston Trio [who recorded "Bay of Mexico" on their debut album, 1958] and the Weavers [who performed it live in 1950-51, later released on Kisses Sweeter than Wine] in the budding "Folk" genre... From there, "Bay of Mexico" (as they called it)...had a bit of a new life, as in the hands of Harry Belafonte for example [who released a "Round the Bay of Mexico" in 1959]. These versions have similar verses (reflecting a more or less common source, the Lomax recording) of a general chantey type and similar to those oft used for "Santiana."...It seems likely that the Folk singers—whoever was first to pick up the song—were exposed to "Round the Mexico" through Alan Lomax's 1941 book, Our Singing Country. (How accessible would Lomax's Bahamas field recording have been at the time?)
Hugill's version is independent of these. And yet, uncharacteristically (but not unheard of) he mentions no source. He does make what appears to be a speculation about an "older hoosier version."