Thanks to a late-niter and with co-credit to the talents of my crackerjack researcher wife, I believe I can trace back "Throw (Put, Place, etc.) A Nickel On The Drum" to its origins as a gospel song.

I'll take it in stages. The first clue came from a songbook privately published in 1928—the earliest version I've so far found in print. It has only a number, no title. You'll notice the lone verse is different—a pointer to gospel origin.

[Throw a Nickel On the Drum
(Salvation Army Song)]

From A Collection of Sea Songs and Ditties of from the Stores of Dave E. Jones, 1928,
http://www.folklore.ms/html/books_and_MSS/1920s/1928ca_a_collection_of_sea_songs_and_ditties__dave_e_jones_(HC)/index.htm

Monday I am happy, Tuesday full of joy,
Wednesday there's a peace within the Devil can't destroy,
Thursday and Friday I'm walking in the light,
Saturday is a heavenly day and so is Sunday night.

Cho:        
[line apparently missing: Hallelujah, hallelujah?]
Throw a nickel on the drum, throw a nickel on the drum,
Hallelujah, hallelujah,
Throw a nickel on the drum and you'll all be saved.

From this came the discoveries that follow (next message).   Bob