One possible way:
I know that optical fibers would be placed inside the nuke (or perhaps the explosive that initiates the nuke reaction). The light pulse would outrun the destruction of the fiber, and arrive at distant instrumentation.
No, the radiation wavefront of soft X-rays advances in front of the visible light (it is what vaporizes the tower guy wires creating the “spikes” in some of these photos). It would destroy the fiber cable before the visible light was transmitted. Radiation also darkens fibers, which leads to it being of limited use in long term radiation environments. Also atmospheric tests were ended well before practical fibers were developed.
I had figured that the detonation was electrically initiated but wondered if the pulse would be fast enough to do that and trigger the camera. They didn’t have diodes but I guess they had capacitors to build delayed signals.
Pretty neat what can be done analog. Many hydraulic circuits have a high degree of programming in them with delays based on pressure pulses as event triggers.