Thanks, Guest. I have collected three other, independent examples of the "Prince of Wales" verses, though one involves the "King of France" and one a "horse." That's enough to show that the words were once fairly well known.
All are sung to the "Hinky Dinky" tune, though yours is the only one with "stinky" in the refrain.
The later George V was Prince of Wales in 1885. (He was 20 years old.) The future Edward VIII, in his early 20s, was Prince during World War I.
If the alleged 1885 reference is correct, the verses (without "Inky...") must antedate WW1. If so, they could easily have been set later to the soldiers' tune, which, esp. in the American army, tended to attract couplets from all over.
I can only guess at the meaning of the verses. "Put in jail/ For riding an ass [or even 'horse'] without a tail" suggests, however, a fairly obvious sexual metaphor.
The "tank" stanza usually has "Three German soldiers" in it. Your version may be the earliest one noted, which suggests an origin in WW2 or even later.
My problem with "Mlle. from Armentieres" is *too much information.* However, I am making progress in assembling it. There's so much good stuff it's difficult to leave so much of it out!
Thanks for asking.