The manuscript I have is from a draft of Christopher Logue's BOOK OF BAWDY BALLADS subsequently published in 1962. He was editing a collection from his old "friend" Count Palmiro Vicarion.

The last verse is clearly of WW II vintage:

Should the Japs make an attack,
Then hundreds of bombs - they will drop 'em,
But we'll keep 'em at bay with an Oerliken gun
Through the hole in the elephant's bottom.

No reference to the King and the Queen or the Strand.

Cheerily,
Charley Noble