On Thursday morning, as Ken Livingstone is on air in London making the first in a series of controversial comments about Israel and Hitler, Jeremy Corbyn is on board a single-carriage train cutting through the Lincolnshire countryside towards Grimsby.

The Labour leader takes off his trademark flat cap and brown satchel to pose for selfies with passengers, some of them brimming with excitement, others bemused.

“He’s the torch carrier,” exclaims one man, flinging his arm around Corbyn, who – oblivious to what was unfolding elsewhere – smiled and cracked a joke. A teenager boasts about the picture of Corbyn he has taken on his phone, before turning and asking, “Who’s he, again?”

At about the time that Labour’s mayoral candidate, Sadiq Khan, tweets that Livingstone’s comments were “appalling and inexcusable”, Corbyn is engaged in an intense conversation with the train inspector about everything from rail policy to the recent catch of a “record cod” to Tony Blair.

In reply, Corbyn reflects on the last conversation he had with Blair before the Iraq war. “Tony, why are we doing this?” he asked. Blair replied: “Because it is the right thing to do.”

Corbyn continues: “I said: ‘That is not an answer.’ He said: ‘That is as good as you’re going to get.” So no good wishes from the former Labour leader? Corbyn laughs. “No, no, I’m waiting,” he says. “It’s in the post.”

A little later, Corbyn is jolted out of this reminiscence and back to the present day. After Livingstone’s fateful appearance on BBC Radio London, the Labour leader’s head of media sees something on his phone. Kevin Slocombe mutters the words that signal the start of one of Corbyn’s hardest days as Labour leader; a day that would result in him suspending one of his closest friends in politics as his party was plunged into a very public row over antisemitism.

“Ken’s said something on the radio,” says Slocombe, a slightly panicked look spilling across his face. “I need a transcript.” Soon the senior adviser slips into the seat beside his leader for a hushed conversation. As Corbyn steps off the train into Grimsby, the darkening in his mood is just perceptible.

In London, Livingstone is striding towards television studios, already defending himself to LBC through a mobile phone pinned to his ear, when he mutters, “I’ve got a violent MP threatening me”.

“You’re a disgusting racist. You’re a lying racist. A Nazi apologist,” spits John Mann, angrily, in an extraordinary confrontation that is caught on camera. “You’re a disgusting Nazi apologist, Livingstone.”

The crowd that greets Corbyn at the Grimsby Institute probably do not notice that he and Slocombe are speaking urgently into their mobile phones, both part of a conference call with other members of the leadership team back in London.

The phone call is aimed at addressing how to tackle a controversy that is quickly turning into a crisis for Labour, as a stream of its MPs join the clamour for Corbyn to take action, and quick.

If the leader himself is panicked, he hides it well. Corbyn hangs up his phone and starts his day, talking to students about the refugee crisis and the “national living wage”, visiting a medical lab, and discussing how Labour should take on the Ukip threat with the local council leader.

Trailed by Slocombe, who stays on the call for over an hour, pacing back and forth as text messages from journalists pour in, Corbyn gathers with activists near the road, some of whom laugh as a passing car honks its horn and the driver hollers: “F**k the Tories!”

Not that he can ignore the spiralling anger being expressed across social media and in television and radio interviews. By lunchtime, Khan’s call for Livingstone to go has been echoed by two dozen MPs, including those in the shadow cabinet, with tough statements from the party’s leader in Scotland, Kezia Dugdale, and Corbyn’s deputy, Tom Watson. In London, the man himself, in retreat from a hungry pack of journalists, is locked in a disabled toilet.

When Slocombe finally comes off the phone, he rushes over to Corbyn’s event manager, Kat Fletcher, who is managing to keep the visit going through all the mounting tension. “I need time with him,” Slocombe says.

As soon as Corbyn emerges from the next event, Fletcher grabs her boss’s arm and gently pushes him through a heavy door in the corner of the room, with the words: “Private meeting.”

But by the time they rush out of those talks, running across a busy road to the town’s cenotaph, where Corbyn is due to lay a wreath to mark workers’ memorial day, a decision has still not been made, and Slocombe is once again lurking in the background on his phone.

The conversation between advisers is a strained one: this inner circle all know that Livingstone has overstepped the mark, but they are also nervous about allowing a narrative that suggests he had somehow allowed antisemitism to flare up within his party.

Corbyn campaigning for local elections in Hull while the storm around Ken Livingstone’s antisemitic comments was breaking. Photograph: Christopher Thomond/The Guardian

Some are spitting fury at Mann, who they feel is simply trying to exploit the situation to destabilise the party leadership in anyway he could.

When the party’s statement finally lands – at 1.24pm – it names both men, suspending Livingstone for bringing Labour into disrepute while ordering Mann to report to the chief whip, Rosie Winterton, for what would ultimately be just a slap on the wrist.

As Corbyn continues with his day of visits, he takes on a dogged manner.

“If there is antisemitism, it has got to go,” he says. “And we have taken action where any issue of antisemitism has arisen. I am absolutely clear in my own mind, I am totally and completely and absolutely against any antisemitism.” Livingstone’s comments are “unacceptable”, he says. He argues that he has campaigned against racism “all my life”.

Still, suspending a close a friend must have been tough? “It is not a happy day,” concedes Corbyn, claiming that the “last thing I want to do is suspend people from the party. It is not my wish to do that.” But Livingstone would have to face an “investigation, and he can put his case”.

This is perhaps Corbyn’s biggest test yet, and would immerse him in 24 hours of relentless media scrutiny – something he admitted earlier in the week to be one of the most difficult aspects of his job.

Sitting in his modest Westminster office, chosen instead of the vast room down the corridor used by his predecessor, Ed Miliband, he throws around adjectives such as “unpleasant” and “absurd” with punctuated emphasis, as he speaks about journalists delving into his family life.

Does he find it hard? “Very hard,” he says, describing journalists investigating “every last aspect” of his home life, “digging up second cousins I’ve never heard of”, and even travelling to Mexico to seek out relatives of his wife, Laura Álvarez, who he claims will probably have never heard of him.

“The Daily Mail used to hide behind that door,” he quips, pointing to the corner of the room, before standing up and taking exaggerated steps towards the door, softening his tread and lowering his voice conspiratorially.

“We were doing shadow cabinet. I told everyone else to keep talking,” he says, gesticulating animatedly behind him. “Then I pulled the door open like this,” he adds, stretching out his arm and yanking it back towards himself, before dissolving into a chuckle. “And they were all there on the ground listening in. I said: ‘If only I had a camera. Why don’t you lot clear off?’”

Corbyn is talking about the night of his shadow cabinet reshuffle, although the Daily Mail was apparently not actually present. Earlier, on Sunday 13 September, four journalists had listened into the discussions between the new Labour leader, aide Simon Fletcher, Winterton and a woman who called herself the “tea girl”.

Corbyn, Labour MP Val Shawcross, Ken Livingstone and comedian Eddie Izzard protesting rail fare increases in 2011. Photograph: Rex/Shutterstock

The reports of what was overheard had Winterton saying that “Andy (Burnham) is in, Hilary (Benn) is in, Angela (Eagle) is in”, but Chris Bryant would have to be out after demanding a 30-minute conversation about what would happen if Britain had to invade Russia.

And that wasn’t the only novel experience for the Islington MP who suddenly found himself at the despatch box for prime minister’s questions.

“It was the first time I’d been on the front bench, ever, in any circumstances … and I looked around and the place was ram-packed, I’d never seen it so full, and I thought: ‘The majority of people here do not wish me well.’”

Six months later and it is clear that Corbyn did not just need to worry about the Conservatives sitting opposite, or the journalists peering down from the House of Commons gallery, but also a number of the Labour MPs lined up on the green benches behind him.

The night of those shadow cabinet discussions was to mark the start of a thorny relationship between the new-style leader and dozens of Labour politicians, on the back and front benches, with some privately admitting that they hope this antisemitism row will be the beginning of the end for a leadership they have never backed.

“I was reading various bits of political history and I discovered that Harold Wilson felt he was very isolated within the PLP,” says Corbyn, drawing a comparison with the former Labour leader.

Does he feel similarly alone?

“Weeellll,” he adds, stretching out the word in hesitation, before plunging into an explanation of how he is dealing with the dissent.

“What Wilson did was to have a Wednesday afternoon open-door policy in his room behind the Speaker’s chair. So every Wednesday afternoon I have open therapy sessions [in the same room] for MPs and they come in, lots of them – I’m not going to name them because it is all very discreet.”

Some of the politicians bring a bottle of whisky in for the 30-minute sessions. Is that to drown their sorrows? Or his? “For me to sign, for raffles,” he says, before talking about how the discussions concern everything from constituency matters to policy, and, he admits, “sometimes they tell me I’ve got it all wrong. I say: ‘Fascinating, explain how you would do it differently.’”

Corbyn on the day he was announced as new Labour leader in September 2015, with fellow candidates Andy Burnham, Yvette Cooper and Liz Kendall. Photograph: Jeff J Mitchell/Getty Images

Asked about what he might do in the face of a leadership challenge – with some MPs talking openly about wanting to take action this summer after the EU referendum – Corbyn turns to Wilson again: “You can spend your whole life worrying about plots, coups and manoeuvres. Wilson’s memoirs are full of this stuff. You can spend a lot of time worrying about it. I don’t. I’ve got a mandate, I’m trying to carry that mandate out.”

But if he is challenged by his own MPs, will it be him taking on the fight?

“We will have an election, of course. I mean, we can stand again. It’s not me. It is ‘we’. We don’t do ‘I’.”

It is a comment that could fuel a rumour already drifting around Westminster that, in the face of a coup, Corbyn – who will turn 71 in 2020 – might prefer his close friend and shadow chancellor, John McDonnell, to take over at the wheel.

Not that he feels too old for an ongoing challenge. “Bernie is eight years older than me,” he says – of US Democratic presidential nominee candidate Bernie Sanders, who Corbyn argues is part of a similar political awakening in the US. “I mean ‘we’ in the sense that I don’t aspire to great personal power or individual ambition. That is not me. That is not what I’m about. We want a strong movement.”

But if that movement takes him to Downing Street, is he ready to be prime minister? Corbyn looks sheepish as he argues that things might be “different” if he were the man in charge. What, so he and Álvarez don’t cherish the idea of packing their bags for No 10?

“I don’t see all that luxury living is exactly me somehow … I like my house,” he says, and his allotment. But he concedes: “You’d probably end up there.”

Yet there are critics within Labour’s parliamentary party who think Corbyn has no interest in taking Labour into power, and would rather head up a protest movement.

While Corbyn admits that he likes the fact that his office gives him a balcony overlooking the Thames, from where he can wave on protest marches as they go past, he insists that he is gunning for a Labour majority.

He claims that to win back seats such as Nuneaton, which was held by the Conservatives last year, requires telling voters that “austerity is not necessary”.

The argument, which suggests that Miliband’s failure was not being leftwing enough, is strongly disputed by a number of MPs. “If people are making a rational political argument, that is fine – let’s have that debate,” says Corbyn; it is the personal attacks he dislikes.

Asked again if he finds it a lonely job, he takes a deep breath. “I’ve been here a long time,” he says. “I’ve been here at times when – for example, the Gulf war, 1991, there were 550 or so who voted one way, and 17 of us voted the other way. I know what it’s like to be in the minority in parliament.”