"This week alone, over 800 refugees were set to make America their new home," the UNHCR said in a statement.
The order signed by Trump on Friday suspends arrivals by refugees in general for 120 days and Syrian refugees indefinitely.
"UNHCR estimates that 20,000 refugees in precarious circumstances might have been resettled to the United States during the 120 days covered by the suspension," the statement added, basing that figure on averages over the last 15 years.
"Refugees are anxious, confused and heartbroken at this suspension," it further said, describing UNHCR chief Filippo Grandi as "deeply worried" by the impact of Trump's move.
Refugees who qualify for resettlement to the US or other developed countries have typically endured extreme hardships and have no prospect of safely returning to their home countries.
The US president's widely-condemned executive order also bars entry for travellers from seven mainly Muslim countries -- Iran, Iraq, Libya, Somalia, Sudan, Syria and Yemen -- for 90 days.
UN human rights chief Zeid bin Ra'ad Zeid al-Hussein, who rarely communicates on Twitter, said in a tweet that "discrimination on nationality alone is forbidden under human rights law", adding that "the US ban is also mean-spirited and wastes resources needed for proper counter-terrorism."
However UN reactions to Trump's order have been largely tepid until now.
The UN bodies most directly engaged with migration -- the UN refugee agency and the International Organization for Migration (IOM) -- issued a statement on Saturday which made no mention of the executive order and stopped far short of condemning it.
IOM spokeman Leonard Doyle on Monday dismissed suggestions that the Trump administration had threatened funding cuts if its policies were publicly criticized.
"I have heard nothing of the sort," Doyle told AFP.
According to IOM's website, the Geneva-based body receives funding from the US Agency of International Development and multiple divisions of the State Department but the details of those contributions were not immediately available.
Doyle said that in the aftermath of Trump's executive action the immediate concern was the protection of refugees whose resettlement process to the United States was ongoing.
"Suddenly they are back where they started or they are back several steps behind," Doyle said.
"We are worried about their immediate vulnerability," he added.
Asked if IOM would push Trump's government to reverse the new policy in the coming weeks, Doyle said, "we are hopeful that in the process of the review, when they have looked at every aspect of refugees, that things will return to an even keel."