As well as being the most popular politician in the country – for now – Andy Burnham is also the most popular man at Westminster. Labour MPs, the unions, Whitehall civil servants, political advisers and thinktanks are all among those battling for the ear of the next prime minister.

“There are so many different demands all at once. But the supply of face time with Andy is significantly smaller than the demand. It’s been like laying down a new train track simultaneously with driving the train at 200mph,” one close ally said.

Burnham arrived back at Westminster on Monday, in a black cab from Euston station, with just a couple of aides at his side. He was given offices on the top floor of the Portcullis House complex, overlooking Big Ben, and on the same corridor as his close allies Louise Haigh and Anneliese Midgley.

In the middle of a heatwave, the two packed rooms have been stifling, his inner circle camped out in one overseeing parallel operations: the leadership campaign, transition planning, policy and communications. But the cohort is tiny: just half a dozen people, with others dipping in.

The pressure on them – Haigh and Midgley but also Sally Jameson, who has been ushering small groups of MPs into the second office to meet Burnham – and other advisers has been intense. Late one night, one went to bed with 450 unread WhatsApp messages.

Miatta Fahnbulleh, a former thinktank chief who quit as a minister over Starmer’s leadership, and Josh Simons, the ex-Makerfield MP who gave up his seat for Burnham, have been drawing up policy for the prime minister presumptive, before a likely coronation in just three weeks’ time.

Team Burnham admit they have been overwhelmed by a “mountain” of policy papers. “Everybody wants to share their ideas. It’s great, but impossible,” one said. They’ve received about 100 submissions since the byelection.

In the thinktank world, policy specialists say they are worried about a “bandwidth problem” because there is too much material and not enough people to sift the ideas. “We have offered help but they don’t seem to be ready for that yet,” said one thinktank boss.

There is a similar concern over the deluge of media requests and queries – his team have been “trying to hold the line”, they said, though at times have struggled as “allies” freelance with their views, sometimes contradictory or just plain wrong.

Burnham himself has so far kept a cool head, but found it hard to hide his blushes when he arrived in Westminster Hall on Monday to see hundreds of cheering Labour MPs assembled for a photograph with him. “Blimey,” he said.

Since then he has kept a lower profile – focusing on private meetings with Labour MPs, many from the 2024 intake whom he is meeting for the first time – in his office or the tea rooms and voting lobbies of the Commons, as he signs up the 81 names required to run for the leadership.

“There’s an absurd amount of pressure on him – but he just takes things in blocks and never gets ahead of himself,” one aide said.

A friend cited the byelection as evidence of his focus. “He could have been distracted by all the national noise, but he prioritised speaking to undecided voters. He wasn’t easily swayed from that, despite all the competing demands.”

But his close allies acknowledge that with so many MPs wanting to pitch for jobs or ideas, it is hard to manage expectations. “He’s just one person and everybody projects their hopes on to him. It’s hard to live up to,” one admitted. “Andy is the personification of this moment of possibility,” another said.

Among Labour MPs there is excitement about Burnham’s charm offensive and anxiety among those who have not yet managed to secure a meeting, amid what one called a “massive bunfight for influence”.

There are also tensions on both wings of the party – with the right taking succour at his appointment of James Purnell, a Blairite former cabinet minister, as his chief of staff, and the left spooked by it. “Everyone projected on to Andy what they wanted, and only some people will get it,” one MP said.

A similar struggle is taking place over Burnham’s choice of chancellor, which more than any other appointment will show his instincts. If he wants to smash through Treasury orthodoxy to pursue his vision of the economy, does he do it with left-leaning Ed Miliband, or a more centrist figure? Will realpolitik prevail over ideological boldness?

The jostling for jobs in a new Burnham administration has been, as one of his inner circle put it, “excruciating”. Every day there is a new rumour, some of it put about by the individual themselves, or their allies.

He has already spoken to some of the current cabinet – including Yvette Cooper and Rachel Reeves – but sources say this was about the handover, rather than potential jobs. Wes Streeting, who rolled in behind his campaign, is also helping.

While Miliband is advising on the economy, so is regularly in touch, the Guardian understands Burnham has not yet sat down with Shabana Mahmood, the home secretary, although he has met John Healey, who quit over military funding. Others are awaiting a call.

There is also talk of “daggers drawn” within some of the outer circles of influence in Burnham’s operation, with some of those who thought they were part of the core team feeling pushed out by Haigh and Simons, who are in the ascendancy. One MP has been almost in tears over it.

Burnham’s team stress that his platform for government is the priority, rather than the individuals who will deliver it. But they acknowledge it is difficult to carve out time.

Back home in the north-west of England this weekend, he is working on his first big speech on the economy – which he’ll make in Manchester on Monday – as well as constituency engagements and using his “stardust and legacy” to help launch Bev Craig’s campaign for mayor.

It is not just politicians and policy wonks who want Burnham’s ear. Keir Starmer has given permission for access talks, but the clock is ticking and senior civil servants are already worried about the short timetable. It will be Purnell’s job to reassure them.

The former cabinet secretary Simon Case said: “Government officials will want and need certainty as soon as possible about who really does speak for Andy. The system needs to know, even in advance of him becoming PM, who has the authority to say ‘Andy would like x’ and really be truly representing his views.”

The demands on Burnham will only intensify when he gets to No 10, although by then he will have an army of civil servants as well as the Labour operation behind him. “This job is not like any of the other jobs you’ve ever done before,” a senior party figure said.

Until that moment – probably 20 July, the first working day after the earliest point he could become leader of the party (17 July) – his team will get by on caffeine and goodwill. “We all have the same dry sense of humour which gets us through. We’re northern realists about everything. We’re taking it all in our stride,” one of them said.

But they are also looking to Burnham himself to get them through. “He knows what he wants to do and why he’s doing it, that really helps guide you. If you’re clear about that, then it helps you prioritise and cut out the noise.”