Assuming you remember the Hiroshima part wrong, you may be thinking of "My Object All Sublime", a 1961 short story by Poul Anderson, which was also the (unaccepted) answer to the question A short story where criminals are sent back in time as punishment?. You might be able to read it at Google Books. Does any of these covers ring a bell?
The story is narrated by the time cop, under cover as a friend of the exiled criminal in 20th century Chicago. The ending:
His excitement softened. He looked from the window and the night, inward, toward the bedrooms. "And my wife and kids," he finished, most gently. "No, I wouldn't go back, no matter what happened."
I took a final breath of my cigar. "You have done rather well."
Liberated from his gray mood, he grinned at me. "You know, I think you believe that yarn."
"Oh, I do." I stubbed out the cigar, rose, and stretched. "The hour is late. We'd better be going."
He didn't notice at once. When he did, he came out of his chair like a big cat. "We?"
"Of course." I drew a nerve gun from my pocket. He stopped in his tracks. "This sort of thing isn't left to chance. We check up. Come along, now."
The blood drained from his face. "No," he mouthed, "no, no, no, you can't, it isn't fair, not to Amalie, the children—"
"That," I told him, "is part of the punishment."
I left him in Damascus, the year before Tamerlane sacked it.
