When Tom Burke was cast in Furiosa: A Mad Max Saga, the prequel to the crash-bang spectacular Mad Max: Fury Road, he sat his 77-year-old mother down in front of the television and showed her the previous film in that post-apocalyptic series, just to give her some idea of what he was letting himself in for. Afterwards, she looked concerned. “Will you be mainly inside or outside?” she asked.

Any parent would worry. As Praetorian Jack, he helps the young Furiosa (Anya Taylor-Joy) take revenge against the pharaoh-like warlord (Chris Hemsworth) who killed her mother. Jack’s job is to sit at the wheel of the War Rig, one of those whopping great tankers without which any Mad Max movie would be underdressed, and shoot high-speed pursuers off their motorbikes. The character is kitted out in battered black leather, not unlike Mel Gibson in the original trilogy, with a smudge of grease across the top third of his face like the mask on a cartoon burglar. In addition to the actor’s own scar from childhood surgery on a cleft lip, which has left a jaunty crimp on the upper right side of his mouth, he sports as Jack a crooked duelling scar under one eye.

Though Burke is saddled with some unwieldy dialogue (“You may be raw but you have about you a purposeful savagery”), he is mostly an impressively grave, still presence in the midst of mayhem. In a key scene, he stands Mad Max-like on a stretch of sun-baked tarmac, while Furiosa points a pistol at him. “I think that’s literally where they shot a bit of Mad Max 2,” Burke tells me. You can hear in his sultry rumble a hint of fanboy giddiness.

It is late morning in a Soho members’ club, and Burke has reserved a snug sofa in a dark corner at the back of the room. The 42-year-old pours himself a cup of green tea with lemon. The Mad Max movies meant a lot to him, he says, even before he got the call from their creator, George Miller, asking him to be in one. “Probably my favourite franchise.” Don’t forget Joanna Hogg’s Souvenir films. “I suppose that is a franchise, yeah.” It was the first Souvenir, in which Burke played a saturnine, upper-class smack addict, that transformed him from jobbing actor to incipient movie star, making good on all his years of promise. Very different, surely, to a Mad Max movie.

“Not necessarily,” he says. “George goes through every beat with you, verbal and non-verbal. Even when there’s an awful lot going on, with cars and flame-throwers and so on, everything still stops before a take, and you do the preamble to get yourself into it. Sometimes that just meant screaming, pretty much. Anya and I did a lot of yelling before a take. That’s not vastly different to what one might do on a Joanna Hogg movie. George knows that without that sense of people affecting each other, things won’t glue together.”

Screaming before a take is hardly the wildest thing he has done. Rehearsing a Bernard Shaw play at the National Theatre, he worked with “a movement chap” who asked him to play a scene as if he were pole-dancing. “Your first thought is: ‘Oh, fuck off!’” he growls. “At the same time, you know it’s probably going to help. You need to be able to take the note that makes you go: ‘Whaaat?’”

Burke is amusing company but has rarely been funny on screen. There was his tetchy jobsworth heavy in the comic thriller Klokkenluider, and his droll improvised asides in The Souvenir, such as when he accuses his girlfriend, who is hogging all the sleeping quarters, of having “bed dysmorphia”. But dourness prevails. I glimpsed him on a street corner last winter: he was glowering from inside a coat which had its … “Was the collar up?” he asks, interrupting me. It was! “I do pop a collar. It’s the first thing I do at a fitting.” It has also been said that he owns quite a few pictures depicting stormy weather. He seems ready to deny this, then tots up the numbers. “Yeah, I do,” he says.

How conscious is he of this sombre persona? “One can easily fall into the slightly aggressive agenda of: ‘I’m a comedian!’” He throws zany viewfinder shapes with his hands, framing his grinning mug as if on a TV screen. “And then you risk losing simplicity and honesty.” He spent some of his 20s going up for wacky Channel 4 series, though it wasn’t only those auditions he flunked. His late godfather, Alan Rickman, who helped put him through Rada, wrote of Burke in a 2009 diary entry: “The respect that he has earned needs to be matched by the work offered.” He smiles at that now, asks me to read it again, then gathers his thoughts. “It wasn’t directors. It was whoever was at the top, ticking off who gets cast in what. There was a decade where I was constantly falling at that hurdle. You think: ‘Oh, I’m the less bankable version of …’” He won’t say whom. “Five names at least.”

His path into the industry, if not through it, has been charmed. His parents, who raised him mainly in Kent, are both in the business: his father, David Burke, played Dr Watson in the first two series of The Adventures of Sherlock Holmes, while his mother, Anna Calder-Marshall, still acts (recent credits include the films Last Christmas and Sweet Sue). But, as Rickman suggested, it took for ever for Burke to gain any ground. As Ryan Gosling’s brother in Only God Forgives, he was dead almost before you’d taken your seat. He was fleetingly in Stephen Frears’s costume drama Chéri with Michelle Pfeiffer, and played an alluringly tangled Dolokhov in the BBC’s War and Peace. But he didn’t find a role that could be both showcase and springboard until five years ago in The Souvenir, executive-produced by Martin Scorsese. “Joanna recognised something original in me. With Scorsese’s involvement, it felt like a tipping point. A change.”

In quick succession, he played Orson Welles in David Fincher’s Mank, then starred with Florence Pugh in The Wonders and Bill Nighy in Living; he also stole Ruth Wilson’s heart (and car) in True Things as a bad boy with a dodgy dye-job. Next up is Steven Soderbergh’s thriller Black Bag, with Cate Blanchett and Regé-Jean Page. He also has a steady TV gig playing a troubled private detective in Strike, adapted from the novels by Robert Galbraith, nom de plume of JK Rowling. It has made him a household name even in homes that wouldn’t know a Mank from a Klokkenluider.

And what would he say to someone who might have avoided Strike because of adverse feelings about its author? “I’m not trying to flog a vacuum cleaner. If you want to watch it, watch it. If you don’t, then don’t.” This sounds snotty, but is said with breezy humility. “I feel complicated about the idea of boycotting, though I’m not saying I’d never do it. I do feel differently about looking at a Picasso or a Modigliani these days. Those thoughts are in my head.”

Burke knows that actors who work with Rowling are expected to declare their affinity with, or antipathy to, her views on trans issues. Yet he has remained fastidiously neutral. “If it’s part of your vocational integrity to say, ‘I’m taking this side’ or ‘I’m taking that side,’ then go for it. I’m not going to stop you. But it’s not mine. I want this to mend. I want everyone to feel safe, and to be neither pathologised nor demonised.” He sounds like a parent asking his kids to play nice. “I don’t have a problem with that archetype,” he says. Some collaborators, though, he would turn down. “I wouldn’t want to work with Lars von Trier. I might be totally wrong but I don’t think I’d feel comfortable in his environment.”

OK, I say, imagine Nigel Farage has written a string of thrillers, which Netflix is turning into a six-parter. Would Burke play the lead. “Oh, come on!” he protests. “This is such a sort of: ‘What if? What if?’” Then he mimes licking his fingertip, and turning the first page of a script. “I don’t know that I wouldn’t have a flick through. I’d be fascinated.

“I see my job as way less important than the meaty stuff of politics and world events. On one level, I’m dressing up. But on another level, I think it’s bigger. And that bigger-ness, that generalness, is sacred to me. That’s why I’m quite protective of what I’m putting out there. Also, I’m fickle. I change my mind.”

Have friends talked to him about working with Rowling? “Yeah! Certainly. I discuss it with them every now and again. And we’re still friends. Look, I don’t think you’re not coming from a place of integrity when you say: ‘I’m not going to watch it.’ And I don’t think Jo’s not coming from a place of integrity. But my place of integrity is what I’m doing.” He looks me square in the eye. “I sleep well at night.”

After such an equable discussion, it’s an oddly blunt end to the matter: a decisive pulling-up of the conversational drawbridge. I think even Burke is slightly taken aback by his own response. There’s a flicker of surprise on his face, which suggests the phrase sounded more reasonable in his head than it did on his lips.

It is nearly time for him to attend a fitting for the Cannes premiere of Furiosa. Before he goes, I ask to see his socks. “Why?” he says, squinting suspiciously. Only because I’ve heard that he prides himself on a strong sock game. “I do!” he says, thrilled that word has got around. Then he looks deflated: “All I’m wearing today is black sports ones.” To think, I went to all the trouble of putting on my best pair so we could have a sock-off. I flash an inch of burnt orange, and he murmurs approvingly. “Mmm. Bright plain. See, that’s usually what I’d be wearing.”

All very jolly. One of the last things he says to me, though, is prompted perhaps by his awareness that an earlier comment might look glib in print. “When I said, ‘I sleep well at night’… Well, you know, that’s just a figure of speech. I do wrestle with all these things.” Satisfied with that, he says goodbye, then darts off to his fitting. Those collars won’t pop themselves.

• Furiosa: A Mad Max Saga is in cinemas from 24 May

• This article was amended on 17 May 2024. An earlier version said that Tom Burke’s father, David Burke, played Dr Watson for a decade in The Adventures of Sherlock Holmes. In fact he was in the first two series, from 1984-1985.