I was looking in the DT and in the Forum for information about a different song with a similar title as this which goes:

This Old Coat of Mine

Now this old cap of mine, the inside is quite new,
The outside has seen some very fine weather,
So I throw this cap of mine to the ocean wild and wide.
(Chorus)
Roll on the boat that takes me home.

repeat for:-
"Now this old coat of mine..."
"Now this old shirt of mine..."
"Now this old vest of mine..."
"Now these old shoes of mine..."
"Now these old socks of mine..."
"Now these old pants of mine..."

As sung by George Collinson of Hull, 1982, and included in the cassette issued by Veteran Tapes "What A Lovely War!" VT121, 1990.

In his book of the same name, Roy Palmer gives the text as "Oh this old hat of mine..." and quotes one soldier's account of a performance of the song:

"[It] was delivered by a singer who gradually stripped off as he sang one verse about each article of clothing and the scenes it had witnessed. When he threw off his pants he was seized by his drunken mates and subjected to various indignities."

Palmer notes this as having been a popular WWII song but which has been reported from earlier on in the 1920's,and probably originated with sailors on the Atlantic packet ships in the mid-nineteenth century, and that a version was cited by Cyril Tawney in his book Grey Funnel Line from a midshipman's logbook of 1860.

As a query - the cassette states that George Collinson's song was recorded by Jim Eldon, while Palmer's book credits Steve Gardham.