As far as I'm aware, the William books are unknown here. When they're referred to in passing in one of the modern-day scenes in Arcadia, 3 of the 6 productions I've seen changed the reference to something similar, as the title is totally unfamiliar in the US.
It's odd about Good King Sauerkraut -- although I grew up on Pogo in the Sunday funnies, I don't remember this bit and Barbara's title made sense to me as a wacky variation on the familiar phrase, even without thinking of Pogo.
I do however remember (most) of the other Pogo carol:
Deck us all with Boston Charlie,
[Line I can't recall]
Nora's freezin' on the trolley,
Polly-wolly cauliflower alley-ga-roo!