The tune played is "Show Me The Way To Go Home" and it is tune number
37, which is stamped into the movement.  The difficulty with the
identification is that the bellows and/or the reservoir are leaking
so badly that the notes are not sustained.

If you are a member of MBSI you can search many articles about
Griesbaum Whistlers, and in particular the 1995 article by Joseph
Schumacher in V41-1, page 25, which lists over 150 titles of whistler
tunes.

Robin Biggins