Rebecca Ferguson is drinking tea in the restaurant of a London hotel. In a dark blue jacket festooned with tiny gold stars, her brunette hair scraped back from her face, she looks perkier than anyone who gave birth six weeks ago has any right to. The 34-year-old Swedish-British actor has been perfectly decent in decent films (such as Florence Foster Jenkins), better than necessary in trashy ones (The Girl on the Train, Life) and has even emerged unscathed from a bomb (The Snowman). But it is her performance as Ilsa Faust, a daredevil secret agent in the new Mission: Impossible film and its predecessor, that has nudged her from the ranks of the interesting to the brink of stardom. Her compelling coolness threatens to outshine her costar, Tom Cruise. Indeed, her opening scenes in the latest instalment, Mission: Impossible – Fallout, and the 2015 Mission: Impossible Rogue Nation, both show her saving his bacon.

“Oh my gosh. You’re right!” she gasps. “I save him both times. I didn’t realise. Though I do remember Chris [McQuarrie, the writer-director] and Tom saying throughout Rogue Nation that every entrance for Ilsa needed to be significant and important. Tom wanted to create an equal for his character, Ethan. He’s bloody cool and strong, but also vulnerable. Ilsa has those qualities, too.” It’s notable that their relationship, while dramatically charged, is powered by mutual fascination and respect, rather than romance. “These are people with a genuine equality between them.”

Ferguson with Tom Cruise, Simon Pegg and Ving Rhames in Mission: Impossible – Fallout.

She can remember the first time she noticed her character’s name, with its baggage. “Faust I understood. But Ilsa looked like a spelling mistake. I thought it was meant to say ‘Lisa’.” She promised herself she would bring it up with McQuarrie. “Then one day, Chris said to me, ‘You know, it’s like Ilsa in Casablanca, right?’ And I said, ‘Yes! I knew that. Of course I knew that.’”

Ferguson isn’t big on theories about why the Mission: Impossible films chime with our times. Their trademark deceptions and switcheroos – where a trusted confidante may at any moment peel off her latex mask to reveal a foe underneath – depict a world in flux. “But if we’d made it 10 years ago, there would have been the financial crisis,” she says. “There’s always chaos of some sort.” Besides, she tends not to focus on anything beyond her involvement or control. “Tom always says he makes the film for the audience, but I don’t. That may be really selfish. I just don’t go in thinking: ‘How will the audience take this?’ I hope they enjoy it, but I do it for myself and the ride – and to make something that I would like to see. I have watched this one three times now.” I ask how it’s holding up and she reaches for her favourite adjectives: “Bloody goddam well.”

Her stillness and composure are a gift to the franchise. Most of the stunts, fights and chases end up playing second fiddle to her bone structure or her wry, unimpressed stare. Did this knack for minimalism come from the modelling she did in her teens? “No, I hated all that. They took us into the audition with little number plates in front of us – cattling us into a prison of prettiness,” she says of her very first modelling job. “After the first shoot, I turned down every job. Even now, I hate being photographed. I laugh my way along the red carpet. Posing makes me stiff and uncomfortable. I look like Bambi on ice.”

Ferguson with Hugh Jackman in The Greatest Showman.

The modelling was partly down to her mother, an adventurous sort with a try-anything attitude. “She once hitchhiked with Sean Connery. He could’ve been my dad! Not that I’m unhappy with my real one.” Her mother is a bit eccentric. “Like a calm version of Edina from Ab Fab. I was more like Saffie. I wish I’d been Patsy.” There’s still time, I say. “You think this is tea?” she replies, raising her cup and slurring her words.

She credits McQuarrie with encouraging her to embrace stillness. “I sometimes find I want to overexplain in my acting. I work a lot with my thought process and I’m always wondering if it’s visible to the viewer. Chris told me a few times: ‘Pull back.’ In one scene, I asked him: ‘What do I think about here?’ He said: ‘Make a goddam lasagne,’ and walked away. So that’s what I did. It was a great way to stop myself overanalysing.”

Understatement doesn’t work in every context. It would have looked perverse amid the kitsch excesses of The Greatest Showman, where Ferguson played the opera singer Jenny Lind. (Her big number, Never Enough, was sung by Loren Allred while Ferguson valiantly lip-synced.) When the bad reviews started coming in for that film at the end of last year, she thought to herself: “Oh, that’s too bad.” She’d had a ball making it. “I never liked musicals before. I always wanted to slap the actors whenever they started singing. Seeing it from the other side changed me.” And she loved working with Hugh Jackman. But these things happen. Films get panned. You move on.

She first realised that wasn’t the end of the story when she heard Zac Efron’s duet with Zendaya from the film in a supermarket in New York. “That’s our music,” she said to herself. The next time it happened, she was back home in the tiny Swedish fishing town of Simrishamn, and they were playing her tune. “That’s when it struck me, the incredible turn this was taking.” Isac, her 11-year-old son, is obsessed with the soundtrack. “He plays it all the time in the car. It’s got to the point where I have imposed a No Greatest Showman month. He’s saying: ‘I want to sing Never Enough,’ and I’m telling him: ‘I can’t take it any more! Enough of Never Enough.’”

In the widely derided The Snowman.

She knows about the sing-along screenings of the film, but didn’t realise they were a regular occurrence at the Prince Charles Cinema in London, 10 minutes from where we are sitting. “No! They’re not still showing it?” A devilish glint flashes in her eye. “I want to go. I’ll sneak in. What if I can’t get tickets? What do they do, stand up and sing? They have costumes? This is too exciting. You know when I take a breath before that line: ‘I’m trying to hold my breath’? Do you think they take a breath, too? But I should go in disguise, right?” She has got herself into quite the tizzy.

What happened to The Greatest Showman – the public reclaiming a film trashed by critics – is the dream, but she has seen the nightmare version, too. Put it this way: the Prince Charles Cinema will freeze over before it holds participatory screenings of The Snowman. Tomas Alfredson, the director of that beleaguered serial-killer movie, was frank about its shortcomings, confessing that 10-15% of the script was never even filmed. “In the end Tomas managed to scrape together what became The Snowman,” she confirms. How does she feel about it now? “I enjoyed it. I know it got torn down. I had a great time working with Michael Fassbender. He does pull-my-finger fart jokes. I mean, come on!” Perhaps there should have been some in the film.

Next she will be seen as Morgana, turning into a dragon in Joe Cornish’s adventure The Kid Who Would Be King. But some of us are still playing catchup with her best work: the BBC’s 10-part Wars of the Roses drama, The White Queen, from 2013, in which she is fiercely affecting as Elizabeth Woodville. I tell her I’m halfway through it and she claps excitedly. “Is it the nipple-less version or the nipple-heavy American one?” I’m embarrassed to confess that I can’t recall any nipples. “I did have discussions where I said: ‘We’re making a very good series and not a cheap porn film, yes?’ I was the lead and I felt I had to be very verbal about that. Great tits, though. Normal tits. I believe in being normal. Get them out there and if they hang a bit – fine.”

Mission: Impossible – Fallout is released on 25 July