The collection "Peter was a Fisherman" (Rounder CD 1114) has a version collected in Toco, Trinidad in 1939. The liner notes say:

Raymond Quevado (Attila the Hun) notes this song as "travelling down the islands from the Bahamas to Barbados and then to Trinidad" [...] (Atilla's Kaiso). The song probably originated in the Commonwealth of the Bahamas during World War I, when cooking oil was difficult to obtain and the majority of cooking was done with local coconut oil. The song entered mainstream US popular music in the early 1930s. Most versions outside the Caribbean appear to be based on US commercial recordings such as Count Basie's [from] 1938. [...] Print versions include one in '"Negro Songs" taken Down by Mrs. Alice Pashley' in Amelia Defries, "The Fortunate Islands" (London: Cecil Palmer, 1929, p. xxi)

The last point in particular shows that the song was in circulation in the West Indies prior to the 1932 recording.