'You've changed," I tell Gillian Anderson. In 1996, she was chosen as the world's sexiest woman by FHM magazine's readers; this Christmas she will be bald and on fire as Miss Havisham in the BBC's adaptation of Great Expectations. So what made her take this role? Anderson bristles: "That's not really a serious question, is it? The real question is, 'How the fuck did I end up as the world's sexiest woman in 1996?' – not why would I do Great Expectations. Any actor would want to do Great Expectations. I never set out to be the world's sexiest woman."

No doubt. In any case, Anderson's beauty is surely more rarefied than that besought by lad mag fans. When Terence Davies cast her in his adaptation of Edith Wharton's novel House of Mirth in 2000, he chose her because her looks reminded him of John Singer Sargent's portraits of American society women.

And when she was cast as FBI special agent Dana Scully in The X-Files, she got the part against the wishes of Fox TV executives. They wanted a buxom, leggy blonde, and got her. "I was hired was because of the insistence of [X-Files creator] Chris Carter that Fox not follow the stereotypical line of casting, and one of the things that he appreciated in me was a certain strength."

We are sitting in a spirit-crushing office in the BBC's soon-to-be abandoned White City complex. It's not quite as decrepit as Satis House, where Miss Havisham spins out her declining years, but it may well become so. It's possibly sexist to say this, but Anderson has whatever she had in 1996 and then some. This is a mother of three who told one interviewer that she gets "Doesn't she have mirrors in her house?" looks on the school run. But today, in white shirt, jeans, stack-heeled shoes, elegantly teased hairdo, and with cheekbones one might celebrate in verse, she is as beautiful as Miss Havisham's ward Estella, though thankfully less cold and cruel.

Anderson is one of two Miss Havishams that the BBC is bringing to the screen during the 200th anniversary celebrations of Charles Dickens's birth. The other, in a film that director Mike Newell started shooting last month, will be played by Helena Bonham Carter. Some claim that Bonham Carter, at 45, is the youngest actor to play Havisham in recent times. Not true: Anderson is 43.

She and Bonham Carter, though, indicate a new trend for younger Miss Havishams. Dickens's notes suggest the character is in her mid-50s. Martita Hunt was 46 when David Lean directed her as Miss Havisham in his 1946 adaptation. Later there was a trend for older actors – Charlotte Rampling or Anne Bancroft in recent times, or Joan Hickson, who played Miss Havisham aged 75.

Anderson has seen none of them. "It's kind of embarrassing that I haven't. But I'm looking forward to seeing Lean's version, and Helena's version of it. I started reading her script and then I thought, 'I'm going to get confused,' so I stopped.

She did, however, see herself as Miss Havisham at a screening earlier in the week. What did she think? "It's just magical. I just felt I want to go up and hug and kiss everybody and tell them what a great job they'd done." What about her performance? "There's plenty of times when I've had a good feeling and then you see the thing and your heart sinks to the bottom of your pelvis. I usually watch, if only just to learn what to never do again."

A friend recently sent her an email saying: "I've always thought of myself as Miss Havisham, it's wonderful that you're getting to play her." But what woman in 2011 could identify with a character whose life stops because she's jilted by a gold digger on her wedding day? "Maybe what they're talking about is their heart being broken 20 years ago and they're still pining. But there is something twistedly romantic about the idea that someone is so in love, that their heart is so broken, that they cannot love again, and they literally stop time.

"There is something titillating and tantalising about pain, whether it's physical pain or our own sorrow or somebody else's pain. If you think about tabloids, the glee they take in somebody else's ruin – there's all of that in Miss Havisham and there's a lot of that in our contemporary existence."

Anderson plays Miss Havisham in a childlike, sing-songy voice. Where did that come from? "When I read a script I hear the voice. If I don't hear the voice, the script's not for me. When I work on something, I work on it in that pitch in my head, but don't actually say it out loud."

Anderson's voice is an unusual instrument. She flits easily from American to English accents, an ease explained by her fugitive biography. She was born in Chicago, raised from the age of three to 11 in London, attended high school and college in the US and started her acting career there before finally returning to London at the age of 35 when The X-Files finished. Go to YouTube and compare her interview on Jay Leno in an American accent with her appearance on Michael Parkinson in an English one. Isn't that odd? "Not at all – I'm an actor," she says.

She is intrigued by other actors who use one distinctive voice for all their roles. "I've just been working with Michael Caine [on a film called Mr Morgan's Last Love] who has got a very distinctive voice. Part of what we love about Michael Caine is his voice. There are some actors, like him, whose voice is part of what we pay for when we see them." What do we pay for when we see Gillian Anderson? "There are people on the planet who ask that question," she replies modestly. "Financiers have been asking for years how much should we pay for whatever she's got."

People started asking that question in 1993 when Anderson was 24 and, against the odds, landed the role of Scully in the then-fledgling Fox Network's globally successful sci-fi drama. By that stage, the drama school graduate had appeared only in two professional stage productions (including an off-Broadway staging of Alan Ayckbourn's Absent Friends with Brenda Blethyn). But it was her performance in an episode of otherwise negligible TV college drama Class of '96 that got her noticed – and got her the role of Scully.

Hadn't Anderson vowed, before The X-Files, that her CV wouldn't be sullied by telly? "That's right. Television back then was really crap. I went to LA to audition for films. I wasn't really reading any TV scripts. Then I read The X-Files script." What was the attraction? "It was a mixture of Silence of the Lambs and something that had never been seen before. Now you see it ad nauseam because everything's copied it, but at the time it was really unique. It was hard not to say, 'OK I'll go in and meet on this', because it was such an interesting, strong, intelligent character."

But surely special agent Dana Scully was always two steps behind Mulder during an investigation? "You are just leaning into the provocative questions!" snaps Anderson. She is possibly cross at going over old, painful ground. "I've talked before about how at the very beginning I was directed to get out of the car after him [Mulder] and go a few steps behind him until I went: 'Hang on a second, what's going on?'"

But what remains slightly vexing about her character was that, after Cagney and Lacey and DI Jane Tennison battled institutional sexism, a seemingly radical series had a woman character who was in some respects a throwback, someone in thrall to a more dramatically interesting guy (plus, Anderson was paid half of David Duchovny's salary – until she complained). The fact that Mulder and Scully seemed to be political liberals only made the structural sexism more unpalatable.

Anderson, however, isn't about to deride the character she played from 1993 to 2002 (with a movie reprise in 2008): "I think that her strength outweighed anything else."

That's not to say Anderson has no regrets about how her twentysomething self dealt with the difficulties of starring in The X Files. "I never had a publicist during the series. I only hired one two or three years ago. When I did, they said: 'What we would have done for you during the series was to separate you from the series, to create an image that was about Gillian Anderson and not about Gillian Anderson as Scully."

Why didn't she have a publicist earlier? "Because I'm a idiot. Once I started paying attention, I started to notice, 'Oh we have a different perception of that person now and that must be the PR machine', which is fascinating." She got more of an insight into that machine when she played Megan Fox's publicist in the 2008 film adaptation of Toby Young's How to Lose Friends and Influence People. "She's trying talks about branding her client." But she wouldn't want to be branded, would she? "No. That's a very American thing. Here the germ is the actor. There the germ is box office, or celebrity, or whatever."

When she quit The X Files and came here to live, was that what she was trying to leave behind? "The only thing I knew was that I needed a break from being on set. I really wanted to do theatre, and for whatever reason, I don't know why, I wanted to be in theatre here rather than in New York." Her 2002 West End debut was in What the Night Is For. Some critics sniffily suggested the audience was composed mostly of X-philes rather than genuine theatregoers.

Why move here when Hollywood was surely beckoning? "All I knew was I wanted to do a play. I had bought a house here already, because I knew I wanted to have part of my life back in London." Why? "'Cause I grew up here, 'cause it felt like home, 'cause I love this city."

By the time she settled in London later in the decade, Anderson had been married twice, first to X-Files assistant art director Clyde Klotz, with whom she had a son, Piper, and then to documentary filmmaker Julian Ozanne. She has had two younger children, Oscar and Felix, with her current partner, British businessman Mark Griffiths.

In the past decade or so, she has compiled a CV less mainstream or branded than, surely, any PR machine would have permitted her. She has appeared in The Last King of Scotland and A Cock and Bull Story. All too rarely has she taken leading film roles – The House of Mirth is an exception. She has been in more classy TV than her pre-X Files self would have thought possible, such as The Crimson Petal and the White and Any Human Heart. She was Bafta- and Emmy-nominated for her Lady Dedlock in Andrew Davies's TV adaptation of Bleak House. On stage she was a well-received Nora in Ibsen's A Doll's House. Yes, she appeared most recently as an MI7 agent in Johnny English Reborn, but let's not spoil the story. We will next see her as an MI5 agent in the Tom Bradby-scripted Shadow Dancer.

What are her acting ambitions? "I want to do everything. I'm an actor." What would she like to be in? "Streetcar." Blanche or Stella? "Blanche." Of course Blanche. She has quietly established a post-Scully roster of women destroyed by brutal societies – Wharton's Lily, Ibsen's Nora, Dickens's Miss Havisham, and, if she has her way, Tennessee Williams's Blanche DuBois. Dana Scully, the role that made Anderson, was in this sense a career blip. "I auditioned for Blanche at drama college. I thought I could have played her then, but didn't – it's tragic. It's something I've always wanted to do before I die."