Bob:
I still feel this may have originated in the UK, the verse:
"Then the army came and they took all my vans
To send to the jungles of Vietnam
Australia's the envy of Uncle Sam
'Cause the Yanks don't have tanks that play Greensleeves."
clearly is post American involvement in the war, and possibly not immediate. However I grew up with ice-cream vans
playing "Greensleeves" as their theme tune throughout late 50's / early 60s.

Nigel