Baroness Warsi, the Conservative former cabinet minister, said it was "the moment we once again lost a little more moral authority", while Sarah Wollaston, the Tory MP, called Mr Trump a "sickening piece of work".
Nadhim Zahawi, Conservative MP for Stratford-on-Avon, said he had been told by a US immigration lawyer that he would be affected by the ban because he was born in Iraq. "A sad, sad day to feel like a second-class citizen. Sad day for the USA," he said.
Binali Yildirim, Turkish prime minister, was openly critical of Mr Trump's refugee ban, saying his own country had done the "sacred and holy" thing by opening its doors to people fleeing for their lives from Syria. "You can't stop this issue by building walls," he said.
There was speculation that athletes including Sir Mo Farah, who was born in Somalia, could also be caught up by the ban. JK Rowling, one of Britain's most successful authors, compared Mrs May to Neville Chamberlain appeasing Hitler before the second world war.
In a statement at 12.05am on Sunday, Mrs May's spokesman repeated the line that immigration in the US was a matter for Washington. But he added that the government would raise the issue with Washington if there was any impact on British citizens. "We are studying this new executive order to see what it means and what the legal effects are, and in particular what the consequences are for UK nationals."
Mr Trump's order, signed on Friday, halts all refugee admissions and has for 90 days prevented people from seven Muslim-majority countries from entering the US: Iran, Iraq, Libya, Somalia, Yemen, Sudan and Syria. He said he needed it to make America safe from "radical Islamic terrorists".
His critics said the order did not cover Saudi Arabia, the homeland of 15 out of the 19 terrorists who carried out the September 11 atrocities.
A US judge has issued a temporary halt to the deportation of immigrants stranded at airports as a result of the order, amid estimates that up to 200 people have been detained at airports or in transit.
The prime minister's trip to the US and Turkey saw her cultivating relations with two of the world's most controversial politicians, Mr Trump and Turkey's president Recep Tayyip Erdogan. In Ankara she endorsed a £100m deal under which Britain's BAE Systems will help develop a new Turkish fighter jet.