When Greta Gerwig was a girl, apocalyptic panic wasn't the done thing. She grew up in northern California, and every spring the Sierra Nevada snowmelt would rush down the mountains into the Sacramento River, which ran so fast and deep she could have knelt on the top of the levee and touched it with her fingertips. Gerwig watched her neighbours load up their cars and drive to the far side of the floodgates, then would ask her father, Gordon, who worked in loans, why their family wasn't doing the same.

"And he always told me, 'If the levee breaks, the whole valley's going under, so we might as well stay put'," she says. "I remember being so impressed by his - what's the word for it? - stick-to-it-ness." Whether a word or not, the Gerwig stick-to-it-ness evidently stuck. At the age of 34, the actress and film-maker has become only the fifth woman in history to be nominated for a Best Director Academy Award (the only one to have won is Kathryn Bigelow in 2009, for The Hurt Locker). She is also the first to be so honoured for her solo directorial debut.

That film - which is set in Sacramento in the early 2000s - is Lady Bird, a coming-of-age story about a square-peg girl in a round-hole town, who dreams of moving to New York City and doing something great. Her name is Christine, and she is played by Saoirse Ronan. Lucas Hedges and Timothee Chalamet appear as suitors, while Laurie Metcalf is her exasperated mother.

The whole thing is so emotionally astute that you can't help but see your own teenage self in it, despite the pinprick specificity of time and place. And among awards voters, it seems to have rung a whole cathedral's worth of bells. Over the past few months, Lady Bird has leapt from indie curio to Oscar front-runner, with Best Actress for Ronan and Best Picture among its five nominations, plus a Golden Globe for Best Film - Musical or Comedy in the bag.

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When I talk to Gerwig on the phone soon after last week's Oscar Nominees Luncheon in Los Angeles, she sounds as if she's bouncing off the walls. Posing for the group picture was "like graduating from the most bananas high school I've ever heard of," she says. "They even announced us as the class of 2017, which was very moving, somehow."

On nominations day, she woke at 3.30am, two hours before the announcement, fretted, then turned her phone off and went back to bed.

"Then I got up at 6.30am and thought, 'okay, whatever has happened has happened'." She had a shower, made coffee, then turned her phone back on. "The first thing I saw was a video Saoirse's best friend had sent me of her crying and laughing at the same time. So I didn't know what it meant, but I thought, 'well, that must be pretty good'."

The first people she called were her parents, who were naturally up to speed. They're her dates for the ceremony, too. Gordon finally has a good reason to pack up the car.

But I first meet her months before any of this craziness, when Lady Bird is the surprise screening at the London Film Festival, and Gerwig's biggest concern is disappointing the audience members who bought tickets on the basis of a Twitter rumour that the surprise film in question was going to be Thor: Ragnarok.

"I hope they're not super bummed out," she says, between mouthfuls of vegetable curry in her London hotel suite. We're talking over a room-service lunch, with Gerwig in "presentable mode" (her words) for a photocall: blue floral tea dress, tousled pixie crop.

In short, she's not the Thor: Ragnarok type. As an actress she didn't break out so much as drift into focus: first as a leading light of the "mumblecore" movement in New York 10 years ago, famous for its semi-improvised, often sexually frank and ultra-low-budget films about hip, shiftless 20-somethings, then more decisively as Ben Stiller's co-star in the 2010 comedy Greenberg - the first of three superb collaborations with the director Noah Baumbach, her partner of the past 10 years. Whatever Lady Bird actually wins or not, the Academy's decision to back it so eagerly hints at a new respect in the business for films made outside the pale, male framework. To be nominated for your debut suggests your peers see a future for Hollywood with you and people like you at the centre of it: between the lines, there's a clear "yes please" to more of this.

Yet her stepping behind the camera wasn't some long-planned tactical manoeuvre. "I didn't write Lady Bird thinking, 'This will be my first solo film,"' she says. "My only initial aim was to see if this story was a movie, period." When she finished the script, then titled Mothers and Daughters, Baumbach offered to direct. "But as soon as I saw it on the page, I knew I had to do it myself."

Despite the clear parallels with her own life - the town, the time, the idea of New York as an Emerald City - the film, Gerwig is keen to stress, isn't an autobiography, veiled or otherwise. "Me as a teenager, I was nothing like that," she says. "I never made anybody call me by a different name, for example." (In the film, Christine takes Lady Bird as her "given name": "It was given by myself to myself.") In contrast, the teenage Gerwig was "a people-pleaser and a rule-follower".

As a child, she was determined to find a way into the arts - first via ballet, then fencing, community theatre and the trumpet, while her parents tried to nudge her towards law school: her mother suggesting entertainment law as a good compromise.

Crumbs of all of the above are scattered throughout Lady Bird - not least the passion for dance but the frankest link is probably the film's depiction of teenagers: they actually look the part, rather than like airbrushed Hollywood dolls. Ronan's character has straggly hair and visible acne - the actress's own, left unconcealed by makeup.

"Movies so often pretend that teenage girls are awkward, yet they never even have a few spots. And the truth about teenage girls is that they do. That doesn't mean they're not beautiful. It just means that's what they look like."

To triangulate the right look, Gerwig and her cinematographer pored over the sun-faded paintings of Sacramento artist Wayne Thiebaud, John Huston's Fat City, filmed in nearby Stockton, and George Lucas's American Graffiti, another teen movie that feels like a documentary snapshot of a moment, and was shot in Modesto, just down the road. Another early favourite was Robert Altman's Hollywood satire The Player: "Because I wanted to work in film and I loved the insider-iness of it, even though I was very far outside at that point. Most California films were about Los Angeles, and my life was a universe away from all of that."

Not any more. Lunch complete, there are pictures to be taken, and an awards-season heavyweight-to-be to introduce to an unsuspecting crowd. It's a sunny day. She stands up and flattens down her dress.

"Okay!" she says. "What's next?"