Like any screenwriter, and a woman besides, Morgan usually wears the cloak of invisibility as surely as Harry Potter once did. But if anyone is casting off that traditional garb it is Morgan. After winning an Emmy last year for The Hour, a 1950s-set television drama about a current affairs programme, the name behind the films Brick Lane, Shame and The Iron Lady began moving up the credits.
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Now, between The Invisible Woman (directed by Ralph Fiennes), Suffragette (the new Carey Mulligan drama out later this year) and her new play, The Mistress Contract, Morgan's stock is set to rise even higher in 2014. Especially now she has traded a room of her own at home for an office. "I'm 45 and for once I feel like I have got a proper job," she says. The only downside is she cannot go to work any more wearing that universal uniform of writers, her pyjamas.
Starring Felicity Jones as Ternan and Fiennes as Dickens, The Invisible Woman is adapted from the biography by Claire Tomalin. The film presents a portrait of Dickens in all his complexity, as a writer, husband, father of 10, celebrity and a man who had his own invisible life. The mix intrigued Morgan.
"He was completely a man of his time who understood and had to live within the confines of social etiquette and sensibilities of the time. His position and reputation were very important to him, and that importance probably grew at the time because there was huge pressure on him, there was more to lose. But then also he was a real visionary. What's fascinating is that for a man who had to conceal so much of himself he was constantly unpeeling the world, the life, the social mores, the social hypocrisy of his time."
Tomalin was happy to comment on the script but even more delighted to leave Morgan to it. Reinventing another writer's material for the screen can be trial and error, says Morgan. "Sometimes it works and sometimes you have moments where you kick yourself for missing out bits of story." She never feels, though, that she is in competition with the writer, seeing the screenplay as a separate creation. "It has to become its own thing, otherwise that's the audiobook."
We meet during the London Film Festival, where Fiennes's second feature as director had its premiere. The location is a Soho hotel, chic but a few notches down the glamour dial from the parties Morgan, a double Bafta winner for the television dramas Sex Traffic and White Girl, attended while in Los Angeles last September at the Emmy Awards.
She had been nominated the year before for The Hour but not won. This time, feeling the pressure was off, she and her husband, the actor Jacob Krichefski, could relax and enjoy the pre-ceremony whirl. "It's an amazing theme park, LA." Winning, when it happened, was "a complete shock".
"When I came off stage it was like the moment where Dorothy arrives at the Emerald City with the Lion and the Tin Man and the Scarecrow. Suddenly you have got make-up and you are walking in front of press and being interviewed. You have this sort of three-minute whirlwind, then you sign a piece of paper, they hand over an Emmy, the real Emmy, then you sit back down in your seat utterly bamboozled."
Another magical thing happened when she went into a party clutching her Emmy. Suddenly everyone wanted to talk to her. Come the second party, she had put the Emmy away and ... nothing. "It's an obvious barometer of what success does to you. You do feel king of the hill but you are reminded very quickly that if you put it down, you're nobody again. It's a good leveller."
By the time she won the Emmy, The Hour had been cancelled by the BBC after two series. I wonder if she saw the award as vindication, or if she had already moved on.
"Over time I found it harder because I have realised there are shows that take time to nurture and build audiences, and I would love to have thought we could have built that audience and carried on." That said, the Emmy was "a lovely full stop". Ultimately it's television, she says. "I'm not saving lives here."
Morgan, a mother of two, was born in Cardiff in 1968, the daughter of an actress and a theatre director. The family moved a lot due to her parents' work, making her adept at learning how to fit in quickly, giving her an ear for accents, and leaving her with a love of putting down roots. "Ironically, the older I get the harder I find change. I love rituals and routines, I love going to the same coffee shop every day and I love the fact my kids are only going to go to two schools."
Morgan hated school, leaving with four GCSEs. She recalls one encouraging teacher and one primary school with affection, but in general there was an "apathy" she disliked: "There was an inherent sense of neglect in those schools."
Her first attempt at writing was "a terrible poem" when she was 15. Eventually making it to university to read English and drama, her first thoughts were of journalism, but she was not keen on the deadline culture. She knew she wanted to write; what she did not know was how long it would take. Ten years of being a waitress and working as a caretaker followed, the latter being particularly useful. "I had very good employers who knew I wanted to write," she says. "I had a computer, a very old PC, and I would write on that every day."
Having done the hard yards, taking writing courses along the way, the most valuable lessons she has learned have come from those working in the business. Courses aside, there is no substitute for sitting down and getting on with it, she says.
"The thing I notice most with writers, the ones who have succeeded are quite simply the ones who have written, they have written beyond the first page. A lot of writers who want to be writers are so overwhelmed with doubt. It is hard. You have to turn that voice off in your head and keep yourself going."
During the early part of her career she got to know Edinburgh well. "For me, Edinburgh will always feel like my twenties," she says. Her husband went to university there, and her plays Splendour, Tiny Dynamite and Fugee were staged at the Traverse. The drama 27, about faith and ageing, was a co-production between the National Theatre Of Scotland and the Royal Lyceum. Her children, 12 and 10, are now at an age when she is looking forward to bringing them to the Fringe.
Films took her away from the stage for a while. Before The Invisible Woman her biggest screen gigs were Shame, with Steve McQueen ("He just raises everybody's game"), and The Iron Lady, a biopic of Mrs Thatcher starring Meryl Streep. The film, I tell her, faced criticism in Scotland for its more personal than political style.
"The expectation and the desire for people to either do a hatchet job on her, or to glorify her, was there, and I think we did neither and I think that was one of the things that was very difficult for an audience," she concedes. "They wanted a much more muscular and political film. I don't think that's the last we will see of films about Margaret Thatcher. I'm sure there are more to come."
We talk some more about The Mistress Contract, her exploration of a relationship that runs until March 22 at the Royal Court, London. From there it is a short stroll back to The Invisible Woman. Does she think it ever works for a woman to be a mistress?
"I've never been a mistress. I've observed mistresses. Any relationship where you are with a man or a woman who is in love with two people is really hard. Ultimately, you are …"
"Somebody at some point is invisible."
Not Morgan, not any longer.
The Invisible Woman opens in cinemas on Friday