This is the only footage that I'm aware of:
I've never seen anything that gave a conclusive answer to this. Part of it is probably that they were recording these things not for general release, for the most part, but as scientific records. The raw audio is not that interesting to a bomb designer, because it doesn't tell you anything about the bomb (whereas the imagery of the fireball, for example, does).
The footage you've linked to (more on that here) was recorded probably by a news organization that was allowed to witness that test in Nevada, which is why it is anomalous.
It would be worth looking into what the difficulties were for recording synchronized audio in these kinds of environments during the period of atmospheric testing (1945-1963). My suspicion is that it raised difficult logistical issues that were solved most simply by just not trying to worry about this extra, extraneous variable. Especially since whenever they wanted to show nuke tests to someone, they would just dub in the generic sound effects.
But again, this is all a hunch. I've never seen any documents that address this issue.
Many of the later atomic bomb tests were filmed by Doc Edgerton from a distance of about seven miles. What you would have had was the flash of the explosion followed by the sound of the blast about 33 seconds later.
The sound would depend on the local weather and acoustic dispersion patterns (at the high pressures, air becomes a dispersive media. High frequencies travel a bit faster which is why thunder has a crack then the rumble.) Any microphones placed closer to the test would be recording a shock wave, and the sound would depend more on the type of microphone than on the type of bomb.
It was just not a high priority to have synchronized sound.
Quite few of these have sound if thats what youre looking for.