The original title by Wilmoth Houdini was "Gin and Coconut Water," copyright 1946.
On the Mitchum album, it was shortened to "Coconut Water." ("That Man, Robert Mitchum, Sings").
It is not included in the Houdini collection on Arhoolie, "Poor but Ambitious 1928-40," with earlier compositions.

There are several old 45s and lps pressed in Jamaica that have the title "Rum and Coco(a)nut Water," unattributed, but essentially remakes of the Wilmoth Houdini song (mentomusic.com). All are post Houdini's copyright date.

Lyr. Add: Gin and Coconut Water
Wilmoth Houdini, 1946

Gin and coconut water (3x)
Cannot get it in America

At the Medical Center what you think
the doctor said you'll have to drink.
He said, "Take this advice from me Houdini"
and go back to the West Indies.

What you need is:

Gin and cocoanut water (3x)
Cannot get in America.

There's one thing that i'm telling you
a catastrophe is happening to me
I want to do the thing that I shouldn't
but when I tried I found that I couldn't.

Gin and cocoanut water (3x)
Cannot get it in America.

Copyright Muziek - Wilmoth Houdini.
www.elyrics.net
This website often has incomplete or error-filled lyrics, so I am not too happy with the lyrics posted above.

I don't have any reliable source for the origin of the changling, "Rum and Cocoanut Water."