It seems there are three different geographical versions. In his book "The phrase that launched 1,000 ships", Nigel Rees tends to believe the "alive and well" began in a natural way, probably after WWII. But (he writes) the phrase was given a tremendous fillip when the Belgian-born songwriter and singer Jacques Brel (1929-1978) made the subject of an Off-broadway musical show called "Jacques Brel Is Alive and Well and Living in Paris". According to Rees, the show was an immense success and ran from 1968 to 1972, therefore paving the way to more and more "alive and well and living..."
By the way, Brel was not living in Paris at the time, but in the Marquises Islands, and he was not well at all.
References to Argentina could be earlier, since it refers obviously to the capture of Adolf Eichmann in Argentina in 1960. According to the Israeli commander of the operation, Isser Har'el, Germany asked Britain to arrest Eichmann in British-controlled Yemen, only to be answered calmly by the Foreign Office that "Eichmann was alive and well and living in Argentina".
There was also persistent rumours that Adolf Hitler was alive and well in Argentina after WWII. That rumor was based on the well-known fact that Nazis had taken refuge in Argentina and that there were a fair number of native Hitler sympathizers there. It was
probably helped along by the practice (common at least in left-wing circles in America) of referring to Juan Peron's regime as fascist.
The third version is from the seventies. The decade began with a religious slogan, 'God is not dead . . . He's alive and well and living in your heart", quickly turned into jokes (Walter Lippman - "God is not dead. He is alive and appearing twice a week in the Washington Post", c. 1970); The famous graffiti "God is not dead, but
alive and well and working on a much less ambitious project" appeared in 1979. But New York was then already queen of that cliché, courtesy of the movie from 1975, "Sheila Devine dead and Living in New York".
For the record, In his preface to "His Last Bow" (1917), Arthur Conan Doyle wrote: "The Friends of Mr. Sherlock Holmes will be glad to learn that
he is still alive and well".