The “Baker” explosion, part of Operation Crossroads, was a nuclear weapon test by the United States military at Bikini Atoll, Micronesia, on July 25, 1946. Photo above was taken from a tower on Bikini Island, 3.5 miles away.
The wider, exterior cloud is actually just a condensation cloud caused by the Wilson chamber effect, and was very brief. There was no classic mushroom cloud rising to the stratosphere, but inside the condensation cloud the top of the water geyser formed a mushroom-like head called the cauliflower, which fell back into the lagoon (compare with this image, a photo taken slightly later, after the condensation cloud had cleared).
The water released by the explosion was highly radioactive and contaminated many of the ships that were set up near it. Some were otherwise undamaged and sent to Hunter’s Point in San Francisco, California, United States, for decontamination. Those which could not be decontaminated were sunk a number of miles off the coast of San Francisco.