Slawko the music man
Other song names: "The German Doctor" "Auch Von music man" "Auch Von de music man" "The German Music Song" "Ach Von De Musica" "Ach Ven de Musica"
The oldest reference I have found to the song is of it being sung by British officers in Canada during WWII (U-Boat Killer - Donald MacIntyre - Chapter 10):
Argentia, [a Canadian seaport in Newfoundland,] so far as we were concerned was an all male society and life often took a fairly noisy and riotous form. [...] Hesperus parties invariably included the singing of a song we had adopted as our own - Lord knows where it originated - known as Zumba Za. It was sung entirely in German and represented a party of people each of whom in turn claimed to be able to play some musical instrument which he proceeded to do in mime, making the appropriate noises and gestures which were then taken up by the assembled party.
It started and ended with a double-bass whose notes were represented by Zumba Zumba Za - hence the name of the song. Somehow this apparently very childish entertainment always succeeded in being in an enormous success, and it was invariably called for when the Hesperus officers were out "on the tiles."
- zum bassa = This sound is used in the "English" versions of the song, including those used in America (source), as part of the faux German "Ich-en-bee-en-zumba-za", which would then be used for other instruments (e.g. "Ich-en-bee-en-viola"). The word splitting is most likely incorrect here, so that it should have been split "Ich-en-bee-en-zum-bassa", where "bassa" is a [double-]bass. The original expression is thus something along the lines of "Ich bin zum bassa" (TODO: German speakers, please try to help turn that into something grammatical, or confirm that this is just faux German), and should be interpreted as "I am playing the [double-]bass".