I haven't found a ton more on the origins of this song, but here are a few about its origin. The earliest references I've found are from 1937. It seems to also go by the name Mango Walk, as attested to here.

" The folk songs of Jamaica have their origin in the distant past, although a few have a distinctly modern note. "Mango Walk" and "Linstead Heath" may some day become popular in America. (link)

The above fragment was reproduced in several papers; the earliest journal carrying it was the St. Louis Globe-Democrat, on May 9th, 1937, on page 14D's article titled "Nail Kegs Used For Jamaica's Drums."

A vague reference from 1924 could be earlier: The Urbana Daily Courier (November 15th 1924, p. 5) in an article about a night of songs "Under Many Flags" which mentions a song titled "Mattie Rag." This makes it seem likely, however they never mentioned Jamaica. (link). Other searches for a song by that name (or other variations on the spelling), however, don't appear until 1943 in The Vancouver Sun (link).

Classical composer Arthur Benjamin wrote a two-part suite called Two Jamaican Pieces composed of two parts: I. "Jamaican Song" and II. "Jamaican Rumba." Composed in 1938, the Rhumba (or Rumba) song is based on this tune. listen (source Boston Globe, June 5th 1955, p. 16B)