Seem to remember Pike doing a version of that, with daft actions , in an episode of "Dad's Army", to keep the rest of crew entertained when they were trapped in a cellar. Stupid Boy!
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V "Vio-vio-vio-la" - as we knew it at school, another rhyme with actions (playing musical instruments), atlhough it is officially called "Music Man", or similar . Also has been sung in modified form as a terrace chant. Not sure if a version was ever sung on the Tivoli, but Norwich fans used to sing "Mick, Mick, Mick, Mick, Mick Maguire", about the Canaries midfielder who later became a PFA official.
There is also "Vintery, Mintery" - an American (?) nonsense rhyme . Mentions "One flew over the cuckoo's nest" in it. Think that's where the film title comes from. Never heard anybody saying / singing the rhyme over here though. (It's on YouTube).
Just checked - can't remember it, but apparently at least part of the rhyme is chanted in the film.