Here's a quote from Straightdope.com. No, I won't vouch for it. I don't know without doing a lot of checking.
"According to the ethnomusicologist Thomas Miller, the song we know (as Kumbaya) began as a Gullah spiritual. Some recordings were made of it in the 1920s, but no doubt it goes back earlier. Published versions began appearing in the 1930s. It's believed an American missionary taught the song to the locals in Angola, where its origins were forgotten. The song was then re-discovered in Angola and brought back here in time for the folksinging revival of the 50s and 60s."
Whoever wrote this did make some mistakes in the article. He says that the Uncle Remus of Joel Chandler Harris is written in Gullah dialect, but that is not true; the stories are written in the normal Negro speech of GA-AL-MS of the period 1890-1930, familiar to everyone who has lived in the rural or small town South.
I think "He's got the whole world.." is a late remake of a spiritual. I will try and check it out.