Theresa May has issued a veiled warning to Donald Trump, arguing that his plan to withdraw from the Paris climate change treaty ranks alongside North Korea’s nuclear missile tests as a threat to global prosperity and security.

In a speech to the United Nations general assembly, the prime minister, whose authority at home has been severely tested since June’s general election result, sought to project her vision of a “rules-based” international order.

She said global cooperation was the only way to confront shared international challenges, including terrorism, climate change, and mass movements of refugees – and condemned countries that fail to play by the rules.

The prime minister did not name the US president directly but made clear that she believed ongoing membership of the Paris climate change accord was as important as the nuclear non-proliferation treaty in maintaining global security.

“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 co-operation 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 she was due to meet Trump for a bilateral meeting 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 on Wednesday night her spokesman said Johnson had changed plans and would fly back to London alongside May on the prime minister’s RAF ­Voyager jet to attend a specially convened meeting of the ­cabinet, where she hopes her colleagues will back her on Brexit.

Her UN address, delivered as she prepares for a major 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.

May also urged the UN to reform, and repeated a pledge by her international development secretary, Priti Patel, that Britain would withhold up to 30% of its £90m a year core funding to the UN, unless it could meet performance targets showing that it was delivering value for money.

“Throughout its history the UN has suffered from a seemingly unbridgeable gap between the nobility of its purposes and the effectiveness of its delivery,” May said.

“When the need for multilateral action has never been greater, the shortcomings of the UN and its institutions risk undermining the confidence of states as members and donors.

“Even more importantly, they risk the confidence and faith of those who rely upon the blue helmets; who rely upon that sign I stand in front of today coming to their aid in the darkest of hours.”

After delivering her speech, May went straight to the bilateral meeting with Trump, at the Lotte New York Palace hotel. Seated in front of US and UK flags, Trump said: “We will be doing a lot of trading with the UK and we look forward to it.”

May said: “We’ve had many discussions between our representatives and ourselves on a whole variety of issues – including trade, which is important for us, and some other policy issues – and our security and defence relationship, which of course is the closest we have.”

The prime minister used the meeting as an opportunity to lobby Trump over the importance of preserving jobs at aerospace giant Bombardier in Northern Ireland, and to press the president over the importance of the international nuclear deal with Iran, which he has threatened to tear up.