My mother used to sing the Rum-tum-tiddle-tum one, but I don't know where she got the tune.
Shula, Christopher Robin Milne (recently deceased) in his later days owned a bookstore in England and was not fond of Pooh references, once calling him "that damned bear". He refused to be photographed with a stuffed Pooh for the article. I gathered from the article that I read that he held some resentment against his father for using him in such a fashion.
I seem to recall reading that Christopher Robin acquitted himself well in the Battle of Britain. Perhaps his comrades teased him once too often. (Here you are fighting for your country in its hour of greatest peril and all you hear is "rum-tum-tiddle-tum".)
I know a gentleman locally who is related to him. He visited him once at his bookstore and confirmed that he touchy on the subject of Pooh.
A bear of little brain,