Whichever definition of "The Government" you want to use doesn't apply to the BBC. It is not controlled by either "The Government" in the UK sense nor is it a government department in the US sense. It is an independent entity that is funded by the public. Your point about funding is entirely...
the Wonderful World of Wicker was actually a spoof of specific tv show call "Wicker's World" and the talking head in the sketch was a parody of Alan Wicker who, apprently, was a big fan of aliteration.