I went to the console and ran GetPlayerControlsDisabled, and saw that the following were all disabled: Movement, Looking, PipBoy, and POV Switch.
Fortunately, I was able to to run EnablePlayerControls, and that appears to have fixed the problem. I haven't seen any ill effects that were obviously caused by it, at any rate.
Don't know what caused the problem. I thought it might be caused by a mod, but I disabled them all and tried again, and still found myself frozen in place after I was given the detonator.
I still have a saved game that I can use to reproduce the problem, so let me know if I can help to diagnose things. I'm just glad that I'm running the game on a PC and have access to the console, and that I remembered seeing the EnablePlayerControls command in a list!