Walking through an airport with Jeremy Corbyn is like travelling with an A-list celebrity. He stops for selfies with 20-somethings on their way to go skiing, waves at excited staff behind duty-free checkout counters and shakes hands with honeymooning couples and OAPs. The only people who don’t appear to be star-struck are confused foreign travellers, trying to figure out who the elderly guy in an anorak and flat cap is. But then, this is a man who was propelled to rock-star status after defying the odds in June’s snap election and had thousands chanting his name at Glastonbury.

Has the adulation gone to his head and got him singing ‘ooooohhhh Jeremy Corbyn’ in the shower? ‘No. And it never will. I’m more likely to sing Born To Be Wild,’ says the Labour leader. We are travelling to Geneva where Corbyn is giving the keynote speech on human rights at the United Nations and discussing Labour’s approach to the greatest challenges facing the world. He’ll also receive an award from the International Peace Bureau for ‘his political work for disarmament and peace’.

That evening, he’s an hour late for our interview and, when he does eventually arrive, he disappears to his room for another 30 minutes – while our photographer panics about missing his flight home because of the delay. When he finally appears in the hotel lounge he’s undemanding and good-humoured. During our photo shoot, the photographer asks him to take off his jacket. ‘Shall I remove my shirt too?’ he jokes, while pretending to unbutton.

He sneers at the chamomile tea in an exaggerated Yorkshire accent: ‘Oh none of that bloody herbal nonsense.’ The photographer works hard to snap a shot of him in a rare moment of silence between the running commentary.

Does he realise, I ask, that his dedication to ‘normcore’ clothing – in the style of cult brand Vetements – has led him to become an accidental fashion icon? How does he rate his style? ‘No, no, no, not that,’ he pleads. ‘I’d rate myself very, very low, OK? Listen, I’ve got prizes three years running for being the worst dressed in Parliament. I dress the way I want to dress. These days people want to dress me differently but I just say no.’

It’s rare to meet a person in Britain without an opinion on Corbyn: whether they think he’s regurgitating the same failed policies from the 1970s, or they are a super-fan of his anti-establishment, anti-austerity views. And while his supporters claimed a ‘victory’ for him in this year’s snap general election (demolishing the Conservatives’ already- slim majority), his detractors point out that he still lost – when he could have trounced a weakened May. Today, pollsters predict he would be the one to seize the keys to Number 10, if the Government were to collapse tomorrow – something that would have seemed fanciful when he was given 200/1 odds of winning the Labour leadership contest in 2015.

This year, Westminster has been rocked by a sexual harassment scandal. As an MP for Islington North for 35 years, he’s part of the parliamentary furniture. Surely he’d heard whispers? ‘Not many whispers, no. I was very shocked,’ he tells me. ‘I’m horrified and appalled by it all. I think sexism is a real challenge in society that needs to be dealt with. The allegations are all investigated and dealt with as appropriate. We support the people making them as well as the people being alleged against.’

As leader of the opposition, he has had to deal with a number of sexual harassment allegations reported to him from within his party. ‘We’re not dealing with huge numbers,’ he adds. ‘We’re dealing with some cases and they are of course disturbing when you get them. . We have a process of dealing with it including a confidential hotline and in a very serious case we appoint an independent person to investigate. And we’ve done that. I’m utterly determined all Labour Party events will be a safe place for women to go to.’

After 30 minutes, his team attempt to call time on our interview. I plead for longer, pointing out that Grazia is the first-ever women’s magazine he has spoken to. Reluctantly, they leave us for another 10 minutes and I ask why he has ignored the Grazia readership in his campaigning up until now. ‘I’ve never been invited to do an interview before,’ he states. That’s not true, I reply, we repeatedly requested an interview with you during the election campaign and kept chasing it for weeks. ‘Well, the campaign was 100 events all around the country. If we could have fit it in we would have, but the demands were huge and the time very limited,’ he retorts. But he did manage to speak to three other magazines (two men’s and one music), I point out. ‘Listen, we have policies including gender pay audits. We have a policy in the Labour Party of women-only shortlists, 61% of all selections made are from women-only shortlists.’ And with that, he won’t be drawn any further.

Since the exit polls came in on the night of the general election in June – where Labour took back an unpredicted 29 seats – Corbyn hasn’t looked back. He thinks ‘there will probably be another election in the next 12 months’ and says he ‘will probably win. I’m ready to be Prime Minister tomorrow.’

He hasn’t (yet) been invited to Prince Harry and Meghan Markle’s wedding but wishes the couple well. ‘She’s clearly a very decent person,’ he adds. I ask what an acceptable amount for the pair to spend on the occasion would be. ‘Weddings come pretty pricey, I understand, but I think the cost should be borne by the family themselves.’ His own wedding to third wife Laura Alvarez in a country hotel in Mexico ‘didn’t cost very much at all’.

With regards to Brexit, does he agree with Tony Blair’s argument for a second referendum: that Leave’s promises during the campaign have now been exposed as untrue, so the country should be allowed to vote again? ‘Some were extremely irresponsible in what they did and said, but we have to recognise it was the largest participation of people in an electoral process ever in Britain and they chose to leave.’ Does that mean he agrees they cast their votes on the basis of false promises, though? ‘People still voted as they did. Yes, I thought there were some ridiculous and exaggerated claims made and I said so at the time.’ So then why isn’t he making it his job to reverse it and bring about a second referendum? ‘I think we should continue putting pressure on the Government to allow a transition period to develop, because at the moment we’re in danger of getting into a complete mess in March 2019 [the date Britain is due to leave the EU] .’

The next day, I shadow Corbyn and his team as they prepare for his speech at the UN. In the talk, he promises action on tax havens and accuses the UK Government of being complicit in violating human rights in Yemen. Beforehand, he’s relaxed and still managing a few jokes here and there. But I want to know whether his jovial good nature is genuine or whether he’s mastered the true politician’s art of charming the crowds. I ask a couple of members of his team – which, interestingly, is mostly female – whether he is always so equable and friendly.