I have heard of - and then later saw on YouTube - a short animated film about the danger of nuclear bombs and their fallout. I read this was first shown on the Ed Sullivan Show about 1953 or so and it scared a number of people not just children. Sounds funny that Sullivan would have aired it with all of the many many live acts he presented, but I'm told he did. Perhaps someone will remember the name of this short film.
I suspect the film you're thinking of was "A Short Vision," made by the British husband-and-wife animators Joan and Peter Foldes.
When I was a little kid, there was a spot or promo film for the United Cerebral Palsy charity that gave me the willies; it showed a puppet dancing until a hand holding a knife or scissors entered the picture and, as the narrator described the symptoms of CP, cut the puppet's strings one by one until it fell "dead." Used to run all the time then too.
For several years I worked in an office ruled over by a clique of Pentecostalist women, who were required by their religion to wear long skirts (denim if I recall) and their hair pulled back in a "bun." (Some of us referred to them as "the bunheads"...behind their backs of course.) They had control over the TV in the lunchroom, and at noon it was The 700 Club or nothing; which they called "the only source for TRUE news." You didn't dare mention to your friends over lunch that you thought, say, Benny Hill was funny; if they overheard you, you were in for a scolding over "watching such filth."
Not long ago I read a comment on You Tube from a woman whose parents banned her from ever watching Jimmy Durante, because they were rabid Republicans and ol' Schnozzola (horrors!!) had backed F.D.R. I can't recall Jimmy's politics ever entering into his act; she sure missed some great entertainment.