With the rise of Jeremy Corbyn, Labour finally has a leader who is in tune with one of its most outspoken leftwing voices. Is Ken Livingstone’s political career on course for a significant new act? “I’m not looking for a job,” he insists, when I visit his northwest London home on a bright morning in half-term. “I’m a house husband. I’ve got to get the kids to school, walk the dog, do the shopping, potter in the garden and be ready for when they come home.”

Livingstone, now 70, is wearing a paint-splattered fleece, faded jeans and leather slippers. Our surroundings also give the impression of placid domesticity. His yellow labrador Coco curls up beside us, opposite a cluster of bikes and scooters in the corner. Guardian supplements coat the kitchen table.

Livingstone sips tea from a Labour “More Police” mug, squinting against the sun streaming in through the skylights. He shares this terraced townhouse with his wife, Emma Beal, who teaches in Hackney, and their two children, Mia, 12, and Tom, 13. He has five children overall in what he has described as “three overlapping families”. His daughter bounces in at one point for lunch, wearing a leopard print onesie.

But the slippers are deceptive. Since Corbyn was elected Labour leader, Livingstone has shot back into the limelight. His moment has come. He chuckles that he’s gone from people in the street asking if he’s “Boris Johnson” or “that ancient wizard in Game of Thrones” to constant media appearances. “Suddenly it’s all whammo! Back!” He has even recently published a memoir-cum-manifesto called Being Red: a Politics for the Future.

Part of Livingstone’s appeal to broadcasters is his position on Labour’s National Executive Committee (NEC). He is involved in its Defence Review, and supports the newly leftwing Labour leadership, which espouses the ideals he held close throughout 14 years as Labour MP for Brent East, five years as leader of the Greater London Council, and eight years as mayor of London.

He tells me that the NEC is working to secure a leftwing future for Labour by changing how the party chooses its leader. When I ask him if a future Corbynite candidate could make the ballot under the current system – which requires a nomination by 15 per cent of MPs – Livingstone smiles. “I think you can rely on Labour’s NEC to resolve that issue.”

He reveals that the NEC is “in the midst” of debating alternative systems, and will “almost certainly” try to change the current process at its autumn conference this year.

“The idea of Labour MPs having a veto over who stands is nonsense,” he says. “The Labour party before Blair was a genuinely open, democratic one. Jeremy’s bringing that back. Exactly how isn’t going to be resolved til the annual conference. But that’s what we want back.”

In his characteristically colourful style, he adds:

“Jeremy is genuinely a democrat. Unlike the New Labour regime, which was more like North Korea, internally.”

Livingstone would like a return to the former process – where a candidate needs one MP to propose them and another to second. But he goes further: “Why not allow councillors to do it, and things like that?”

This plan will alarm moderate MPs, who already fear being purged. Indeed, Livingstone warns: “The only area where I disagree with Jeremy and John [McDonnell, shadow chancellor] is I do think we should have automatic reselection all the time.”

Why? “Those Labour MPs were there during a Labour regime that never reversed Thatcher’s anti-union laws, never gave people the stronger work rights that we had in the past – yet they demand them for themselves. It’s wrong.”

Livingstone, an example to the new Labour leadership of a lefty who can win elections, was central to the party’s socialist set in the Eighties. Many of the same figures from that era are back. He worked closely with McDonnell at the GLC, his long-time strategist Simon Fletcher ran Corbyn’s leadership campaign, and he and Corbyn were allies in the Commons. They even planned to run for the leadership against Blair:

“The Socialist Campaign Group met in ‘94 after John Smith died and we kicked round the idea of me running for leader, and Jeremy for deputy,” Livingstone recalls. “But we could only get about 13 or 15 votes. So we never had a chance.”

Isn’t it time for a new leftwing vanguard, without the baggage and the vulnerability to being labelled “dinosaurs”?

“Really, starting with Kennedy in 1960, running right the way through Blair, there’s been this obsession with youth and attractive, personable young men – and it is invariably men,” he replies. “The mess we’re in now, though, I think people are looking for someone who’s older and understands something, and knows how to run something.”

Not that Corbyn has had a great deal of experience beyond the backbenches. And there are a few areas where Livingstone disagrees with him. He calls it a “mistake” that Corbyn did not sing the national anthem at a Battle of Britain anniversary service, shortly after he was elected leader.

He also disagrees with the Trident compromise hinted at by Corbyn and his shadow defence secretary Emily Thornberry – to build the submarines, but with no nuclear missiles. “No,” he says. “I’d use the money to rebuild our military and our neighbourhood policing – that’s crucial.”

When I ask if it will be impossible for Labour to reach an agreement on its nuclear stance, Livingstone replies: “Well, of course!”

Livingstone thinks that two factors could lead to a Corbyn victory in 2020. The first is economic difficulty: “People are really struggling, there’s been no improvement, except for a small elite,” he says. “If we’re really lucky, things will just continue to be painful. If we’re really unlucky, there'll be another crash. That’s the road to what’s coming. That’s the potential. If we can convince people there’s a better way, we can win.

The second is Corbyn’s personality: “I’m hoping [the Tory leader will be] Osborne, because he’s the most obnoxious snob in British politics. That’d be good. The contrast between George Osborne and Jeremy Corbyn, who everybody is going to accept is actually a nicer and more regular guy, is exactly what we want.”

So while Livingstone may claim to be semi-retired, he is building a late legacy: a future for Labour defined by the politics of his past.

Being Red: A Politics for the Future by Ken Livingstone is out on 20 February (Pluto Press and Left Book Club)