With most political polls ahead of the presidential election pointing to a Hillary Clinton victory, the news media spent weeks wondering if Donald Trump would stick to his word and refuse to concede if he lost the election. In the wee hours of Wednesday morning, TV anchors were faced with a different question.

“I didn’t ask Hillary Clinton. I obviously made an oversight,” Fox News anchor Chris Wallace said less than an hour before the network called the election for Mr. Trump.

In...