UNITED NATIONS — António Guterres took the oath of office on Monday to become the next secretary general of the United Nations amid a rise in nationalist movements around the world and what he called a loss of confidence in institutions, including the one he will take over in January.

The next United Nations leader would already have faced tough challenges: war, climate change, widening income inequality, record levels of global displacement. But the election of Donald J. Trump as president of the United States has changed the incoming secretary general’s approach to virtually every major crisis, a wide array of United Nations diplomats said in recent weeks, outlining three particularly vexing conundrums.

The first is that even as he needs commitment from the United States, the single largest funder of the United Nations, Mr. Guterres will be under pressure to call out American leaders if they flout the basic values of the United Nations Charter, including “respect, human rights, tolerance and solidarity,” as he described them to the packed General Assembly hall on Monday.

“The threats to these values are most often based on fear,” he said. “Our duty to the people we serve is to work together to move from fear of each other to trust in each other. Trust in the values that bind us, and trust in the institutions that serve and protect us.”