At least one episode of the wartime sitcom It's That Man Again (ITMA) featured an ambassador from an unnamed European country's government-in-exile whose dialogue consisted entirely of the word "Dvorak" (like the composer). This was, of course, translated into perfectly comprehensible English by his interpreter.
The Goon Show's Little Jim always spoke in gibberish except for his Catchphrase: "He's fallen in the wa-ter!" This was lampshaded in the one-off revival show The Last Goon Show of All.
Little Jim: (Gibberish) Bluebottle: Eccles, I do not understand what he is saying. Eccles: Just a minute, I'll ask him. What did you say, Little Jim? Little Jim: (Gibberish) Eccles: He said that he doesn't understand what he's saying either. Bluebottle: Oh. He's one of Mrs. Thatcher's incomprehensives. [A reference to Britain's controversial Comprehensive School system].
Michael Palin, talking about how influential the show's Absurdity Ascendant principle was to Monty Python and other British comedy/culture, said that when his father walked in while he was listening to The Goon Show he thought the radio was broken.
Dead Ringers did this with their parodies of real-life Labour ministers John Prescott and Robin Cook (which, though exaggerated, was somewhat Truth in Television):
John Prescott: I think you'll find that in point of fact that I make no apology for the fact and thank you you've had your say and now I'll have yours as well years of Tory neglect and the Prime Minister agrees with me on this and I think you'll find that the Government's position has been consistently that and! Robin Cook: Hvee shaw hvay hmm hmm ethcal furn pulsie?
Bob & Ray would occasionally feature agricultural reports from the consummately unintelligible Dean Archer Armstead, of the "Lackawanna Field Station."
The Boston-based Howie Carr Show has an occasional feature in which he plays recordings of Mayor Thomas Menino speeches and invites listeners to try to figure out what His Honor is saying.
In the Australian comedy How Green Was My Cactus King Bonza (a riff on then Australian Prime Minister Bob Hawke) evades the questioning of an interviewer well known for his gravelly speech by offering him a lolly to chew on, then deliberately misunderstanding his now unintelligible questions.