I just want to clarify which nuke you're talking about. There was the first bomb, nicknamed "Gadget", that was detonated at the Trinity test. Then there was Little Boy and Fat Man, that were detonated at Hiroshima and Nagasaki respectively. Gadget and Fat Man were both plutonium implosion devices, while Little Boy was a uranium gun type weapon, gun referring to how the bomb was triggered internally. There was a fourth design called Thin Man that was a plutonium gun type weapon, but it's development was cancelled due to various design issues.

I suspect the bomb you are referring to is commonly known as the "Demon Core." This has quite the interesting history behind it, and I highly recommend looking more into it, but the long and short of it is that it was involved in 2 criticality experiments that went wrong and killed 2 scientists, and gave high doses of radiation to many others. /u/restricteddata, an accomplished nuclear historian, who frequents this subreddit, contacted a former Los Alamos scientist, Glenn McDuff, about what happened to the final core. It was apparently melted down and used in other cores. Before this it was long suspected to have been used at Bikini Atoll in post-war testing.