There's no mention of this song in the Mudcat threads or database. Here's what I know about it:

Fantail Fanny

Fantail Fanny, the girl with the quadruple screw…

Notes:

This was composed by an officer aboard the Light Cruiser USS Columbia for a musical show during World War 2, in memory of a visit to Sydney, Australia, from remarks by Captain Charlie Nelson, USN (Ret.), November 24, 2003.

Not to be confused with a nightclub song of the same title: Fanlight Fanny-1935C
Decca F5569 Gifford/Cliffe.

I'm tried some of the bawdy song websites but found nothing so far. I would appreciate any help.

Charlie Ipcar