X + Y adds up to success for Paddington star Sally Hawkins

It's been a remarkable 12 months for Hawkins. A year ago, she was an awards season contender - Bafta, Golden Globe and Oscar nominations for her work in Woody Allen's Blue Jasmine. Then, as if to prove she had box-office clout, she landed in two of the year's biggest hits. Admittedly, audiences flocked to Godzilla to see a city-stomping monster rather than her scientist, but being a part of anything that grossed $528 million worldwide is always good for the career. "I never thought I'd get it," she laughs.

More surprising was the utterly delightful Paddington, with Hawkins charming as Mrs Brown, the quirky matriarch who insists her family look after Peru's most famous bear. So far it's taken $196 million around the globe; and Hawkins got to work with her all-time hero, Julie Walters. "She's always been a huge inspiration," she nods "[for] her comic ability, her intelligence and her phenomenal acting ability - right back to Acorn Antiques."

If citing a national treasure as your idol may suggest Hawkins isn't exactly edgy, it's entirely in keeping for an actress who is as grounded as they come. We first met for Mike Leigh's Happy-Go-Lucky, her breakthrough performance that won her Berlin Film Festival's Best Actress and a Golden Globe. I asked then if she ever wears designer clothes. "I don't really," she replied. "Top Shop and H&M!" Even at last year's Oscars, where Hawkins looked stunning in a pearly Valentino gown, she had an almost apologetic look on her face.

Today, she's in character - jeans, a purple blouse, tights and wedge-shoes - though you could easily imagine this ensemble came from Hawkins's own wardrobe. Pale and fragile-looking, she has long lashes of dark hair which she uses "as a kind of helmet", a sort of makeshift disguise for when she's in public. With her privacy precious to her, Hawkins is not the sort you'll find flaunting herself on random red carpets. "This world of celebrity can get wrapped up in so much stuff that's not important," she says. "The work is the important thing - not who you are, the actor."

It's this sort of quest to keep it real that's fed into Hawkins's best work - films like 2010's Made In Dagenham, in which she played the Ford factory worker fighting for equal pay. X+Y falls into the same category. Directed by first-time feature director Morgan Matthews, it's inspired by his own 2007 documentary Beautiful Young Minds, which followed a group of young British maths maestros as they competed for a place at the International Mathematical Olympiad while struggling with various levels of autism or Asperger's.

In the film, Butterfield plays Nathan, similarly bright with numbers and "on the spectrum" of autism, says Hawkins, who plays his mother Julie. While the story follows Nathan to Taipei, as he tries out for the British team competing at the IMO, it's his relationship with his mother that provides the very heart of the film. Closed-off, distant, cold even, Nathan is an emotion-free zone. "The fact that she doesn't get much back from him," says Hawkins, "is quite hard for her to accept and deal with."

While much of this has to do with the fact Nathan's father was killed in a car crash years earlier - glimpsed in a visceral opening prologue - Hawkins admits it was the subject of autism that intrigued her. "I've just always been interested in it, weirdly, just on a personal level. I just find the way people think and how stress affects it and how there are various degrees and who is on the spectrum and who is not..." She tails off, unsure of her words. "Especially in the creative field, you come across a lot of people who... I think we're all a bit... I suppose."

Hawkins flew in from Vancouver, where she was shooting Godzilla, and only had one day to prepare. But at least she's had a friendly face on set - Rafe Spall, who plays Nathan's maths teacher. Just a few months prior, they'd been working together on stage on Nick Payne's play Constellations; so did Spall twist the arm of X+Y's director to get her on board? "I don't think so! I hope not!" she cries, looking aghast at the very idea. Spall is unequivocal when talking about Hawkins. "I think she's amazing - one of the best actresses in the world. She's my friend, she's my dear friend."

It's no surprise they're so tight - with Spall also from Southeast London and hard-wired into the Mike Leigh family, via his actor-father Timothy. Hawkins was born in Lewisham, and raised in Greenwich and Blackheath. She grew up with an older brother, who now runs his own web-design company, and her parents, Jacqui and Colin Hawkins, who were both teachers before deciding to write and illustrate children's books together - including the brilliantly named Zug The Bug.

Encouraged by her parents to be creative, Hawkins's earliest memories were "making crowns of wire and creating paper cut outs of dolls' houses", she says. "They were probably awful, but they never said!" She loved drawing and painting. "I suppose I was copying what they were doing," she muses, though it wasn't all arts and crafts. Her childhood home was full to the brim of books by everyone from Roald Dahl to Maurice Sendak; and it was her parents' love of children's lit that "first started to inspire me to read".

Bookish and insular, Hawkins was shy as a child. "I know a lot of actors say that... but I'd take everything on myself and worry a lot as a child. I love to laugh, and I did laugh a lot, but alongside that was a real shyness and insecurity." As a way to counteract this, she gravitated towards performing. "I was always interested in acting from an early age. It was either art or drama. I didn't know whether to go to art school or drama school, but I followed that path."

Plumping for RADA, she won a place at the first attempt. "It's got a great name and that opens doors," she says, briefly sounding calculating. "For me, there wasn't anywhere else to go." Graduating in 1998, Hawkins's first screen appearance was in an episode of Casualty, but her first film role was far more prestigious - playing the sexually rapacious Samantha in Leigh's 2002 film, All Or Nothing. Two years later, they reunited on his abortion drama Vera Drake, Hawkins playing Susan, the upper-class girl who is date raped.

Leigh reports that when he first met Hawkins, they bonded over the work of artist Ronald Searle, the creator of St Trinian's, and there's something of the mischievous schoolgirl in Poppy, the bubbly primary-level teacher Hawkins eventually played for him in Happy-Go-Lucky. Even the character's bright orange underwear belonged to Hawkins. "I do have a similar energy to Poppy," she grins. "I naturally have an optimistic approach to life and I'm told I'm a smiley person. I'm a giggler."

This isn't hard to believe; Hawkins is full of smiles. But she's a wary soul in front of the press. She's vague about where she lives - a flat in southwest London is about as much as you'll get - and any questions about her personal life would doubtless cause such red-faced embarrassment that I don't have it in me to delve into her dating history. While she seems so rooted in London life, she could quite easily uproot to the US if the right offer came her way, she says.

Recently, she shot a TV pilot How And Why, written and directed by Being John Malkovich scribe Charlie Kaufman. A typically surreal idea about a children's television show that acts as a portal to a supernatural world, it didn't get picked up - though if it had, Hawkins would've been packing her suitcase for the States. Instant stardom, though, is not something she's seeking. "I'd rather do good work and keep working," she says. "Just as quickly as you can become the next best thing, you can quickly not."

X+Y screens at GFT on February 25 and Grosvenor on February 26 as part of the Glasgow Film Festival. It goes on general release on March 13