I'm not nuclear weapons expert, but I'm under an impression nuclear warheads (or bombs) have a pretty precise mechanism of detonation. Blowing it up externally wouldn't thus start the nuclear chain reaction, it would just cause an ordinary explosion and spread some radioactive waste (a dirty bomb). This is partly the reason why it was so hard to build a nuke to begin with and why a portion of scientists claimed it's not even possible.
It would have been better if the object had been under a massive barrage of ordinary weapons fire, thus rendering it unable to detect the nuke in time.
You can see quite a few kilometers away on the sea. Following your idea the object would have logically destroyed the nuke so far away it would have suffered no consequences whatsoever.
I didn't realise how young the the MCs were before checking AniDB. I was assuming they were adults and basically at least had graduated from high school already, possibly had some extra education under their belts as well. I mean, I don't see why they would send an uneducated kid to serve as an object engineer. It can't be that easy (and actually it isn't, as we saw when he accidentally tortured the pilot with the seat belts). The only conclusion is what Shinta mentioned: Contrary to total wars having been ended with the objects, it's actually a much shittier world than ours (at large), where even underage civilians are forced to serve in the military because there aren't sufficient amounts of adults left anymore.