‘Gentlemen, you can’t fight in here, this is the War Room!” In fact, it is a rehearsal studio on an autumn afternoon, but let’s not quibble. This is where Steve Coogan and the rest of the cast are running through a new stage version of Stanley Kubrick’s 1964 satire Dr Strangelove. Watching over them is their director Sean Foley and his co-adapter, Armando Iannucci, who together resemble a tall, gruff bricklayer and his short, smiley mate.

The actors pace around in front of an as yet unpainted wooden set, which revolves to reveal the office of General Ripper, played on screen by the peremptory Sterling Hayden and here by John Hopkins. As the scene unfolds, Ripper barks down the phone at the stiff-upper-lipped Group Captain Lionel Mandrake, played by Coogan, who is feebly clutching a computer printout in front of his face.

It is Ripper’s mental breakdown that sets in motion a cataclysmic chain of events beginning with the launching of US missiles at the Soviet Union. The rest of this abyss-black comedy follows the efforts to intercept or otherwise thwart these apparently unstoppable nukes as Armageddon looms. Columbia Pictures, thrilled at the sight of Peter Sellers donning various disguises in Kubrick’s 1962 adaptation of Lolita, stipulated that the actor should assume multiple roles in the film. He played Mandrake, the wheedling US president Merkin Muffley, and the sinister nuclear boffin Dr Strangelove. He was also lined up for a fourth role – Major Kong, the B-52 pilot eventually played by Slim Pickens, and memorably seen riding a warhead as though it were a bucking bronco – but an ankle injury put paid to that.

Coogan, who has frequently been compared to Sellers, and narrowly missed out on playing the Pink Panther star in a 2004 biopic (Geoffrey Rush pipped him to the post), is going one better than his idol and taking on all four parts. Among the photographs pinned to the studio wall today is one showing him in a Stetson, riding the missile as Major Kong. “Steve playing all those roles will add a bravura quality to the story,” says Foley as he and Iannucci take their seats in this basement room in London that is not unlike a nuclear bunker.

There is the added complication that some of Coogan’s characters appear together in the same scene. Muffley and Dr Strangelove, for instance, are both in the War Room simultaneously. “Yeah,” Foley says, sucking the air through his teeth. “Could you tell us how to do that?”

A minor detail. What matters is that the sensibility of the original should be intact. As the novelist Kazuo Ishiguro has written, the movie is “a remarkable blend of things that shouldn’t go together … Kubrick created the darkest vision you could have of nuclear war and combined it with out-and-out comedy.”

That wasn’t how it started. Kubrick was adapting Peter George’s suspense novel Red Alert when it struck him that the extremity of its doomsday scenario demanded to be treated irreverently. The Cuban missile crisis had only recently abated when he enlisted Terry Southern, author of the comic novel The Magic Christian (which Sellers had given to Kubrick), to tease out the funny side of the impending apocalypse.

Selling it as a laugh riot presented another sort of challenge. Mo Rothman, the head of Columbia Pictures, told Kubrick: “The publicity department is having a hard time getting a handle on how to promote a comedy about the destruction of the planet.” Some critics found the movie too strange to love. Bosley Crowther in the New York Times called it “malefic and sick”. Says Foley now: “We can only hope.”

Any shock value has surely diminished. “I doubt the audience will be saying, ‘You shouldn’t make jokes about that!’” says Iannucci. “Hopefully the play is a reminder of the stakes involved in all this. We look on people like Trump as a sort of entertainment, whereas the decisions they make in power have global consequences. It’s not just 30-second clips on TikTok.”

Both men harbour a lifelong love of Kubrick’s movie. Iannucci was born two months before it opened. “So he wasn’t at the premiere,” Foley points out. “I first saw it on video and loved it,” says Iannucci, who knows a thing or two himself about the comedy of brinkmanship, having created the political sitcoms Veep and The Thick of It, and directed The Death of Stalin.

He also co-created The Day Today, the 1990s news parody in which Chris Morris’s Paxman-esque presenter escalates an innocuous item about a peace treaty into a declaration of global war. That episode is pure Dr Strangelove, as is the scene from In the Loop, the cinematic spin-off of The Thick of It, in which a US general works out the viability of a potential war using a pink talking calculator.

“The American release of In the Loop had a Dr Strangelove-type poster,” Iannucci recalls. “And the French producers of The Death of Stalin told me they thought it had a Strangelove-esque feel. I’ve always admired things like Brazil or The Great Dictator: that idea of themes that are so immense that we can only respond in an absurdist way. Take the scene in Dr Strangelove where Mandrake needs to call the president to stop World War III but he has run out of change. That’s a great comedy situation.”

Foley believes Dr Strangelove to be Kubrick’s masterpiece. “I’m always interested in taking something that’s brilliant and doing it in another medium,” he explains. “Though it can be a poisoned chalice because people might say you’ve ruined it.” So far, so good: Foley’s stage versions of The Ladykillers (in 2011) and Withnail and I (with which he ended his tenure earlier this year as artistic director of Birmingham Rep) were both greeted warmly.

“Ideally, people will leave not quite knowing which bits were new and which were from the film,” says Iannucci. “When the Kubrick estate was looking over the script, there were lines where they had to go back and check whether they were in the film or not. That’s a good sign.”

Some elements were crying out to be overhauled. “The film is a satire on maleness and power as well as politics and the arms race,” says Foley. “There’s only one real female character, though: a bikini-clad secretary who’s having an affair with one of the generals.” Iannucci winces: “Different times,” he says.

“We didn’t think there was any way of staging that today,” Foley continues, “but then that would make it a show with only men in it.” A solution arose in the unlikely form of Vera Lynn, whose rendition of We’ll Meet Again provides the film with its bitterly ironic punchline, and who now appears as a character, played by Penny Ashmore. “She gets dropped on Russia,” jokes Iannucci.

As if anyone needed reminding that the real world is scarcely more stable than the one depicted on stage, the US election will take place early in the run, and is sure to provide an extra frisson. “The worse the world gets, the better it is for the play,” Foley says.

Even without that, there is the spectre of Elon Musk, who, in his propensity for making the planet a more volatile place, could easily have stepped out of the world of Dr Strangelove. “You can have fun at Musk’s expense,” says Iannucci. “But I find it menacing that people in charge of information are prioritising rumours and lies that conform to their point of view.” In recent months, he has been needling Musk whenever possible for the benefit of his almost 700,000 followers on X. “Call it Twitter!” he says, thumping a fist on the table in mock fury. He denies he is mounting any kind of sustained campaign of irritation against the platform’s owner. “I’m just sitting down of an evening, seeing him on there and going, ‘Ah, fuck it.’”

On the day we meet, Taylor Swift has just come out in support of Kamala Harris, signing her endorsement statement: “Childless cat lady.” Musk responded by tweeting: “Fine Taylor … you win … I will give you a child …” Iannucci shakes his head in dismay. “You just go, ‘What? Are you threatening to impregnate someone? Just because you run this company, you feel you can write whatever you want?’”

I ask whether he’ll be putting aside a couple of tickets for Musk on opening night.

“There must be a restricted view,” he says. “Let’s give him that.”