BBC correspondents in Europe, Russia, Ukraine and Asia reflect on what world leaders make of a Burnham premiership.

US: Burnham has opportunity to win over Trump

So far, Donald Trump has been fairly dismissive when it comes to Burnham. He seems unconvinced that a Burnham premiership will address his two biggest criticisms of Starmer's government: its policies on immigration and drilling in the North Sea oil field.

US administration officials, for instance, view the possibility of Ed Miliband being named chancellor of the exchequer as evidence that British environmental policy will remain unchanged, given his ban on new North Sea oil and gas drilling licences as energy secretary.

Sir Keir and Trump had a surprisingly warm relationship from the start of the American president's second term. For now, Trump seems to know little of Burnham beyond his being a "mayor of a town" and suspecting that he is "extremely liberal".

Beneath the surface, however, US officials are already pressuring the Burnham team to form a government to their liking. While they have the aforementioned concerns about Miliband, they are hoping for continuity in British foreign policy leadership, which includes keeping Yvette Cooper on as foreign secretary

Although the UK under Starmer has boosted its defence spending, Trump – and top US security officials – want the British to set an even higher example for the rest of Europe. The tensions over US access to British military bases during the Iran War are still simmering as well.

Burnham being a relative unknown in the US poses both risk and opportunity for him once he moves into 10 Downing Street.

The opportunity is that, with a proper charm offensive, Burnham - like Starmer - could win Trump over in spite of their political differences.

The risk is that, given political pressures in both countries, those differences – on issues like military spending, digital service taxes, energy extraction, international conflicts and trade - could end up defining their relationship well before a personal connection has a chance to form.

China: A sense of frustration at more political change

Burnham once pointed to China's high-speed rail network as a model for the North of England.

In 2018, he appeared open to what China could offer at a regional level. As prime minister, the calculation is more complicated. Any relationship with Beijing now has to be judged against both economic benefit and national security risks as British intelligence services continue to raise concerns about spying, threats to Chinese nationals in the UK and intellectual property theft.

Starmer took a broadly pragmatic approach, seeking to balance those competing interests and becoming the first UK prime minister in eight years to visit Beijing. The question is whether Burnham will follow suit - or take a different line.

If Burnham returns to China, he will likely also be shown the country's latest infrastructure advances - a reminder of its economic ambition and also its political continuity.

Chinese officials will be watching closely. They gave Starmer a tour of the Forbidden City and a lavish banquet as part of a worldwide charm offensive in the hope that the UK and others will look at Beijing as a stable predictable partner in contrast to the US.

However, there is a sense of frustration in Beijing at the frequency of political change in London.

Some Chinese analysts see the turnover of UK prime ministers as a sign of instability, and what Zhang Jian, from China Institutes of Contemporary International Relations, calls the country's "identity crisis" since Brexit.

Russia: Moscow expects no change in UK relations

In Moscow the UK is often described as the Kremlin's "enemy No1". In part that is because of Britain's longstanding military and financial backing for Kyiv.

The UK is also the subject of Kremlin ire simply because relations between the US and Russia have improved so much during Trump's second term.

Russian newspapers believe they have the measure of the incoming PM. The government publication, Rossiyskaya Gazeta, describes Burnham as "a consistent critic of Russia, who constantly calls for the West to adopt a tougher stance on Russia".

On social media last month Burnham reminded his followers he had opposed Russia's invasion of Crimea in 2014, opposed Russia hosting the football world cup in 2018, and had supported Ukraine and its mayors since 2022.

The Kremlin does not expect a change in Downing Street will bring any change in British policy. President Vladimir Putin's spokesman, Dmitry Peskov, says Starmer "hasn't distinguished himself" in terms of Russia-UK relations and accuses the outgoing prime minister of "maintaining these relations at the very lowest level".

As for Burnham, Peskov is frank: "It's unlikely that anyone on the British political scene will have a different position to Keir Starmer on our bilateral relations."

That in part is because UK support for Ukraine - and Russian hostility towards the UK - has remained remarkably constant throughout Britain's recent political turmoil.

Ukraine: Concern in Kyiv about UK political instability

In the fifth year of its full-scale war with Russia, Ukraine is about to see its fifth UK prime minister.

There is some concern in Kyiv about the risk of political instability in Britain. Ukraine needs its allies to remain focused on the giant war in the heart of Europe; it needs the UK to keep up its military aid, its drone deliveries and its economic support for "as long as it takes", as it promised.

But from PM No 1, Boris Johnson – so popular he had a cake named after him here - right through to No 4, Keir Starmer, British backing for Ukraine has been constant.

There's no sense that would change under Burnham given his past comments.

More broadly, his foreign policy priorities aren't clear. But from Ukraine's perspective, it's important for the new PM not to waver in recognising the threat that Russia presents to the West and to remain committed to helping Ukraine defend itself, and Europe, accordingly.

EU: Europeans anxiously watching UK defence spending

The view across the Channel is that the UK's newfound political volatility simply means they've joined the club. Governments in Germany, Denmark, France and more are pretty stormy affairs these days too.

Practically, EU countries are curious to see if Burnham harbours the same fervour Starmer claimed to have for a "reset" of economic relations and whether the new UK PM will be willing to budge on Labour's red lines to date: a no to freedom of movement, rejoining the EU customs union or single market.

On defence, European allies are anxiously watching to see if the UK will honour its Nato commitment to hugely increase spending to 3.5% of GDP by 2035.

Burnham appears to have clear domestic priorities. In the volatile world we live in, with threats from China and Russia, the needs of Ukraine and an often hostile US president, Europe's overriding question is - will the new UK prime minister prioritise continental security too?

France: Paris hopes Burnham's world view is like Starmer's

Burnham has spent little of his political career dealing with the world outside the UK, so logically the outside world's knowledge of Burnham is also strictly limited.

That is certainly the case in France, where the foreign intentions of Britain's likely next PM seem like a closed book.

In the only interview any official has given on the matter, Foreign Minister Jean-Noël Barrot wished Burnham well and hoped for "as much stability as possible" on the other side of the Channel.

The Paris government will have been told that Burnham, at least in a loose kind of way, is pro-European, but that he is very deliberately not advocating a return to the EU. Not that different from Starmer then, with whom the process of "re-setting" relations with the EU was already under way.

The French government will want more of that, as well as more co-operation over stopping the "small boats".

And more too of the unspectacular but steady deepening of defence ties. The Starmer-Macron team has proved effective in co-ordinating on Ukraine and the Strait of Hormuz.

India: Delhi will be most concerned with continuity

Burnham is mostly an unknown quantity in India. But as one career diplomat told me, Delhi wasn't particularly "familiar" with the revolving cast of prime ministers over the past decade either.

That churn is at complete odds with the Bharatiya Janata Party's (BJP) firm grip on Indian politics, its own leader Narendra Modi in power now for more than a decade. Whether that's a strength or a weakness depends largely on your political leanings in India.

For now, officials have stayed silent publicly. The assumption in Delhi is that the UK's political saga may not be over. A former Indian High Commissioner to the UK, Yashvardhan Kumar Sinha, has described Burnham as a "popular leader" facing "a very resurgent Reform".

Delhi will be most concerned with continuity, with a focus on policy not personality. There is a broad view that the UK-India relationship is on an "upward trajectory", India's former foreign secretary Shyam Saran told me.

Top of the agenda will be keeping the UK-India free trade agreement, due to be implemented in July, on track. The government will also be watching for any potential changes to immigration rules, particularly for Indian students and skilled workers. (BBC)

Update news