“A special key of some sort.”— In-game description
The nuke key is a miscellaneous item in Fallout.
This is the key of nuclear warheads control console, which is located at 4th level of Los Angeles Vault under the Cathedral.
This key can be obtained from the Lieutenant's locker at 4th level of Mariposa Military Base and then used at a console to nuke the Master. It sets up the nuke bomb timer.