AUDIO...I remember years ago -about 1965 - playing with my tape recorder. A friend of mine showed me his little joke: "Hey, Paddy, [Irishman] whatever you do, don't dump this stuff here!" "Don't do what?" "Dump this stuff here!" [Silly voice:] "OK" - followed by recording of breath blowing into the microphone, but slowed down to 1/2 or 1/4. The white noise was converted into low frequency noise, sounding like a long rumble or explosion. A slowed-up noise sounds ponderous and heavy. The film Time Bandits by the Monty Python American, Terry Gilliam (was any American ever called Terence? Really??) has a giant; his feet hit the sand with a slow thump. Generally, slow sound gives the effect of slow, and therefore far off, rumbling explosion.
VIDEO...OK - the designers of atoms bombs couldn't do simple bright flashes - too difficult. And the 'mushroom cloud' was forced on them. So, they filmed explosions and their sooty clouds, whatever would work - maybe far off, but, to save money; more likely closer with a faked background. A middle distance explosion with hydrocarbons would produce a nice bang, and smoke-ring type clouds. When slowed up, these would look like a huge 'mushroom cloud', slowly churning internally.
AUDIO AND VIDEO...ALL THAT'S NEEDED IS TO PLAYBACK IN SLOW MOTION --- TO MAKE IT LOOK LIKE A HUGE, REMOTE, EXPLOSION - WHICH COULD BE PASSED OFF AS NUCLEAR. I'd suggest the number of supposed nuke tests, where the sound arrives far too soon, are just slowed-up explosions designed to produce smoke, where the film-makers didn't bother to change the sound-track to simulate the delay caused by the relatively slow sound waves.
I'll try to post a Youtube to show how it's done.
[NB there's a different effect with some supposed underwater 'nukes' - where a whole surrounding ring of explosives must have been used to make the central explosion look larger. And of course other effects where clouds/ condensation/ whatever it's meant to be are projected or added directly to the films.]