Donald Trump has asserted his relationship with the UK is “more up” with Andy Burnham than under Keir Starmer despite tensions between the two countries on a range of thorny issues including regulation of artificial intelligence, the Chagos islands and the war in Iran.

At their first face-to-face meeting, the US president praised Burnham, who sat nervously alongside him, as a “natural businessperson”, saying he thought he had “lots of assets on his side” and would be a “great” prime minister.

Burnham, who was joined in the talks by Ed Miliband, the foreign secretary, and Jonathan Powell, his national security adviser, said he and Trump had built on the “good connection” they established early on, in an effort to shore up relations which had splintered over Iran.

UK officials reacted with relief that the talks appeared to have gone according to plan. But despite the warm words at the UN general assembly in New York, there were still signs of difficulties ahead.

On his way to the US, Burnham had revealed he would urge the US president to de-escalate the Iran conflict, with turmoil in the Middle East pushing up global energy prices, putting the UK economy under intense pressure.

Much of the summit has been dominated by geopolitical turbulence, with Trump using his speech to double down on his strategy. The president repeated his threat to “annihilate” Iran, adding he would “drive them into hell” if they did not back down and sign a peace deal with the US.

Asked later whether he had any sympathy with allies which were feeling pain over the cost of living as a result of his actions, Trump claimed that “as soon as the war is over the prices are going to come way down”.

Trump, who has suggested Iran could agree to a deal after the US midterms in November, said “people understand” the need to prevent Tehran obtaining a nuclear weapon, even if that meant price hikes at the petrol pumps and in shops.

In his own UN speech, Burnham was expected to push for greater international control over AI. He revealed to business leaders that he wants a “single set of global principles and standards” to guard against rogue AI, saying the issue would be a main focus of the UK’s G20 presidency next year.

But Burnham’s ambition for the UK to be an “honest broker” between countries in that could set him on a collision course with Trump, who has called warnings about AI’s existential threat to humanity a “hoax” and gave a defiant speech in which he intensified his criticism of attempts to regulate the emerging technology.

“The United States also totally rejects any attempt to construct a globalist scheme to control for the artificial intelligence being spoken of so much now. We’re going to encourage it, not rein it in,” he said.

“Americans have never been a nation that retreats from a frontier or shrinks from a challenge, no matter how great or how daunting that challenge may be … I’m ⁠not going to stifle growth of something that will be bigger than ‌the Industrial Revolution.”

In another schism between the two countries, Trump was scathing about the UK’s deal to give away sovereignty of the Chagos Islands to Mauritius, while paying for a lease of the military base of Diego Garcia.

“I don’t support it. I think it’s terrible. It was a terrible deal. [Mauritius] all of a sudden say they have ownership. Somebody has ownership that never showed up before after decades and centuries. I think it’s ridiculous,” he said.

But he suggested that Burnham may reconsider the deal. “It’s amazing how strategic it is, and I think the prime minister is going to be looking at it,” he said, to which the UK leader nodded and said “yeah”.

However, Trump avoided commenting on the UK’s decision to ban trade with West Bank settlements, leaving it to the prime minister to answer the question. He is also understood to have agreed to a request from Burnham to bring up the case of imprisoned Hong Kong dissident Jimmy Lai in talks with China’s Xi Jinping on Thursday. British officials in the meeting suggested Lai could live in the UK.

UK officials had been somewhat nervous about the meeting, as Trump had previously dismissed Burnham as “extremely liberal” and described the former mayor of Greater Manchester as the “mayor of a town”. However, his demeanour towards Burnham was mostly cordial.

But it appeared to have gone to plan for both sides, with Trump praising US-UK relations, saying: “I think you’re up. Well, they’re more now than they were with your last prime minister, let me put it that way.

“I think right now, based on my relationship with your prime minister, Andy, I think we’re doing very well.”

Later in the press conference, Trump hit back at reporters questioning why he has banned some journalists from the White House, leading to a wider boycott by media organisations. The president claimed he had a “right to clean out fake news”.

In his UN speech, Trump promised there would be peace between Ukraine and Russia “more quickly than people understand”. After talks with Ukraine’s president, Volodymyr Zelenskyy, he warned that Starmer’s rhetoric had been “too tough” towards Moscow and that London “needs to be a little bit careful”.

Many of Trump’s talking points were contradicted, however, by the UN secretary-general António Guterres, who used his speech to warn about the consequences of international conflict, climate change and AI over the past decade.

“Geopolitical divides have deepened. Inequalities have intensified. Trust has frayed. The climate crisis went from distant warning to daily reality. And artificial intelligence moved at a speed that has blindsided even those who created it,” he said.

Without naming Israel, he added that what followed the Hamas attack of October 2023 “was an onslaught on Palestinians in Gaza, with a scale of killing and destruction unlike anything I have witnessed in all my years as secretary-general”.