Richard Burgon had always been interested in politics, but it was only in the grand surroundings of St John’s College, Cambridge, that he realised he wanted to be an activist.

The Labour MP for Leeds East did his English Literature degree there and chaired the university’s Labour club. But it wasn’t academic work or student politics that fed his political aspirations: it was his fellow students.

“They were the kind of people that [I] until then wouldn’t have thought existed apart from in a Harry Enfield sketch,” he chuckles, sitting across a round wooden table from me in his airy Westminster office. He wears a smart suit and purple floral patterned tie. The Labour party founder Keir Hardie addressing a crowd in Trafalgar Square roars above us on a black and white poster hanging on the otherwise sparse wall.

“The thing I took with me from there was that the people at Cambridge were not cleverer than the people I went to school with, in an inner-city school in Leeds,” he says. “I realised not only are they well-connected, not only are they likely to be running the country or be in influential positions power, but actually they’re not that impressive either. So why should we leave it to them?”

Burgon, 37, was raised by two teachers in Leeds, where he used to organise rock festivals; he was – and still is – a dedicated metalhead. Aged eight, he heard an Iron Maiden song on a friend’s Walkman during a holiday to Malta. He immediately bought a bootleg tape and hasn’t looked back since. “It’s a form of escapism,” he says.

Although he was the first in his family to go to university, the world of politics was a little more familiar. Both his parents were Labour members, and his uncle Colin Burgon was a Labour MP, representing the since-abolished west Yorkshire constituency of Elmet from 1997-2010. A teenage Richard would occasionally make trips down to London to visit him in the Commons.

“Obviously I grew up hearing about politics,” he tells me. “And hearing about, for example, the miners’ strike, because my auntie was married to a striking miner, and he was on strike for the whole 1984-85 miners’ strike.”

Burgon did his dissertation on the poem “V” by Tony Harrison, set in a Leeds graveyard during that strike, and went straight back to his home city after graduating to become a trade union lawyer. It’s a route that set him up for his current job as shadow justice secretary, which he landed in June 2016, having served as shadow city minister since Jeremy Corbyn became leader.

Elected in May 2015, having been active in his local party while practising law for over a decade, Burgon’s arrival in the Commons came shortly before his party began moving in his preferred political direction.

Since nominating Corbyn in the first leadership election, he has been a loyal ally, his socialist credentials shaking the business community somewhat when he was in the shadow Treasury team (the FT called him “Jeremy Corbyn’s reluctant man in the City”, and he famously couldn’t say how big the deficit was in a Channel 4 interview shortly after he was appointed).

Burgon’s commitment to Corbynism has led to speculation that he could be a potential successor. He admits that “it’s interesting to speculate, obviously” on who could follow Corbyn, but calls it a “political soap opera”, and thinks “Jeremy’s going to be leader for a long time”.

Nevertheless, he seems positive about what the next leadership contest – whenever it is – will look like. “There’s plenty of talented people in the parliamentary Labour party from different political strands actually as well,” he says. “So that’s encouraging. And it could be, who knows, that the next leader of the Labour party’s not even an MP yet.”

Burgon is looking to redefine justice as a pillar of the welfare state, rather than something people cannot relate to until they need it. “Sometimes people, when they imagine justice, imagine the high court, supreme court, judges in wigs,” he says.

“I want to be making the point that when we’re talking about legal aid and access to justice, we’re talking about people being able to assert their rights against rogue landlords, people being able to get justice in relation to abusive or violent husbands, people being mistreated by unscrupulous employers.”

This means Labour reversing many of the government’s legal aid cuts, including “family law, benefits law and housing law as a starter”, Burgon says. He claims that this would cost around £50m.

Having represented people in Leeds who had been unfairly dismissed, or discriminated against at work, as a lawyer, Burgon would also like to give the profession a bit of a rebrand. He’s keen to “stress the value – not only economically but socially – of lawyers… and not just in the pro-bono work they do but also in the bread-and-butter stuff they do as well”.

Burgon is also rethinking private contractors running and maintaining prisons. “I think it’s morally wrong for people to profit from the incarceration of human beings. But of course there will be some role for the private sector,” he reveals.

“For example, let’s think about hospitals. Some of the examples that have been built more recently don’t have the laundries in them, they’re elsewhere. There are certain practicalities you may have where a company doing the laundry in a prison or a hospital,” he says. “So it’s not an absolutist position. But we don’t think the private sector provision of maintenance in prisons is working.”

We speak before Jeremy Corbyn’s speech on Labour committing to a permanent customs union with the EU, and I ask Burgon how he feels about 87 per cent Labour members being in favour of staying in the single market, and 85 per cent thinking we should stay in the customs union, according to a party members survey by Queen Mary University. Should Labour give them more of a say?

“Motions of this sort can be brought to conference,” Burgon says. “And I know the leadership listens to what the membership says on this and a whole host of issues.”

So would the leadership go by what members voted for in such a motion, or would they have to ignore it? “It depends on the nature of the vote,” he replies. “It’s not me in charge of that.”

Burgon believes Labour should prioritise “what is best for the country… But also what is in the interest of showing Labour believes in the democratic process”. He feels his party “showed real pragmatism and maturity in not falling for the demand that we just ignore the referendum”, warning against trying to reverse the result.

“My fear would be that we would be falling into a trap that the extreme right would like us to fall into,” he says. “People like Nick Griffin [the former British National Party leader], also people like Nigel Farage, would have secretly been praying the Labour party came out the next day saying ‘guess what – that doesn’t count’, and I think that could lead to a right-wing populism taking hold.”

So, as a committed heavy metal fan, which band or artist does Burgon most liken the current Labour party to? He gives a small embarrassed grin. “The Labour party is so cutting-edge at the moment, and I’m so out-of-touch with the latest music scene! It’s hard to find a comparison.” He pauses, and then inspiration flashes across his face: “We’re not just playing the greatest hits; we’re playing new tunes as well.”