“As the global system struggles to adapt, we are confronted by states deliberately flouting – for their own gain – the rules and standards that have secured our collective prosperity and security,” she said.

She singled out Russia, Syria and North Korea for direct condemnation, but described the climate change treaty as part of the “rules-based system” that protects global peace and security.

“It is the fundamental values that we share, values of fairness, justice and human rights, that have created the common cause between nations to act together in our shared interest and form the multilateral system.

“And it is this rules-based system which we have developed – including the institutions, the international frameworks of free and fair trade, agreements such as the Paris climate change accord, and laws and conventions like the non-proliferation treaty – which enables the global cooperation through which we can protect those values.”

She warned that undermining these international institutions, including the UN, ultimately threatened states’ national interests.

“If this system we have created is found no longer to be capable of meeting the challenges of our time, then there will be a crisis of faith in multilateralism and global cooperation that will damage the interests of all our peoples.”

The prime minister was the first world leader to visit Trump in the White House, brushing off concerns in some European capitals about his unpredictability.

But she has repeatedly expressed concern about his decision to seek to renegotiate the Paris treaty, and a planned state visit to the UK has been indefinitely postponed, amid fears of public protests.

She delivered the speech just an hour before meeting Trump in a New York hotel. In his own speech to the UN general assembly on Tuesday, the US president described the North Korean leader, Kim Jong-un, as “rocket man”, and threatened to “totally destroy” Kim’s country if it did not rein in its nuclear ambitions.

May issued her own strongly worded condemnation of North Korea’s actions, but stopped short of threatening military action.

“Time after time he has shown contempt for the international community of law-abiding states,” she said of Kim Jong-un. “Contempt for his neighbours. And contempt for the institutions and rules that have preserved peace and security.

“On this challenge the UN has in recent weeks shown it can step up to the task, with last Monday’s Security Council resolution creating the biggest sanctions package of the 21st century. We have seen regional and global powers coming together and – as in its founding charter – putting aside limited self-interest to show leadership on behalf of the wider world.”

She also criticised Russia for using its veto on the UN Security Council to block tougher action against Syria for using chemical weapons on civilians.

“One country in particular has used its veto as many times in the last five years as in the whole of the second half of the cold war. And in so doing they have prevented action against a despicable regime that has murdered its own people with chemical weapons,” she said.

May was applauded when she recounted the terrorist attacks in London and Manchester earlier this year, and said: “The terrorists did not win, for we will never let anyone destroy our way of life.”

The prime minister’s trip to New York has been overshadowed by speculation about Boris Johnson’s future, after the foreign secretary wrote a 4,000-word article for The Daily Telegraph that appeared to set out a distinctive vision for Brexit.

But her UN address, delivered as she prepares for a speech on Brexit in Florence on Friday, was aimed at presenting Britain as a free-trading, outward-looking country that would continue to play its role in the world.