The following link will take you to p 58 of "Gunner's Tales and Memories of WW2" by Caroline J Eddleston on which The Winkle Song appears. The footnote says the version presented was written by six serving soldiers in 1940.

https://books.google.ca/books?id=UO7xAgAAQBAJ&pg=PA58&lpg=PA58&dq=I+CAN%27T+GET+MY+WINKLE+OUT&source=bl&ots=HMiRlATIp9&sig=M74KZ