The actress Sofie Grabol, who was Sarah Lund and is now shortly to be Queen Margaret of Scotland, is teaching me, in a rather amused way, how to correctly pronounce the word Forbrydelsen. In a sun-filled office at Birmingham Rep Theatre, I haltingly try to produce the word while asking her about her most famous role. In Danish, Forbrydelsen (I said "For-brill-son", she says "Fur-bruld-sen") means The Crime. In the UK, the series was called The Killing. Lund, of course, was the character that introduced the UK to Grabol: a focused, stern, rigid, armed, be-jumpered, coffee-devouring detective with long hair, little patience, a stifled air of melancholic angst and a predilection for becoming involved in gruesome murders.

By contrast Grabol, after politely correcting my diction, is short-haired, smiling, disarmingly warm, sparky and appealingly garrulous. We meet as she begins technical rehearsals for her key role in The True Mirror, the third of the National Theatre of Scotland's James Plays, the historical plays written by Rona Munro which will be premiered at this year's Edinburgh International Festival (EIF), and indeed could be said to form the dramatic centre of this year's festival.

She is looking forward to spending nearly a month in Edinburgh for the role, although her two children, 13 and 10, will not come with her and she will miss them. She hopes to visit Stirling Castle, where Queen Margaret (who lived from 1456 to 1486) once resided. Margaret was a Danish princess who was transported to Scotland at the age of 12 to marry the mercurial King James III of Scotland. It is clear Grabol is in love with both the character and the script written by Munro.

Loading article content

But before medieval history, we tackle modern Danish history - and how her downbeat, very Danish detective series became a cult classic in the UK. And although Grabol's mind is firmly on The True Mirror, directed by the NTS artistic director Laurie Sansom, she is delighted to discuss The Killing. Pecking at a chicken salad, dressed in a cardigan, black trousers and livid green trainers, she speaks of the series as if it was an unexplained phenomenon that swept into her life and left leaving only elation and slight bewilderment.

Grabol, 45, had been a stage and TV actress in Denmark since she was 17 but was unknown outside her home country until the series, conceived by Søren Sveistrup, crossed the North Sea, with subtitles, to such success in 2011. For most of the audiences in Edinburgh this month, the series will be the only show in which the actress has been seen.

Grabol speaks with a lilting, strongly accented but immaculate English. "I know Sarah Lund is well known, but I am not. I don't get recognised. If she walked down the street with her long hair and ponytail and jumper maybe … but no one knows me." She nods and adds: "I know some people have only seen my acting in the The Killing. I am not only happy about it, I am very proud about it. Maybe people will think of Sarah Lund when I enter the stage and think 'when is she going to pull a gun?' or 'where's her jumper?' Maybe she stands in the way but I don't think so."

Lund, who appeared in two more series of The Killing (Sveistrup wanted her to die, but the series ended with her flying away to an uncertain fate), is a beloved character but essentially just another role in the actress's long career. "Personally, it is so much in my bones to move on," says Grabol, the daughter of two architects. "I have been acting since I was 17. And the beauty of this job is entering new projects, meeting new people, and telling a story whether on film, TV or stage - and then moving on.

''To me there is no mourning in moving on. It's more that the success of The Killing is almost fictional to me - I still don't understand that it was such a success." She shakes her head. "I still really don't understand it. It's incredible. And the loveliest thing is that it is a very local, very Danish story, the whole storytelling - it's about a flat structure and not being very passionate. Everything being held back. It's the rational way and the understated way."

In her year away from Denmark, she says she had realised "when you meet something very different from you, is when you somehow define yourself." She says Munro's play shows the Scots as "passionate, uncontrollable" with the Danes as "rational, logical". She believes that view of Denmark is "spot on".

"The Danes do not get lost in our feelings. We are rational and/or boring. Nothing crazy every happens in Denmark. As a Dane, when we look at your [Westminster] parliament and see the way they sit and scream, it's like a play - it would never happen in Denmark! To go on with all that silly passion."

But you became an actress, I say, that involves passion. "Becoming an actress is not rational?" She laughs heartily. "I am not saying the Danes cannot be creative. And that is something else that is very inspiring for me as a Dane, is that the Danes, and I, definitely have this slight embarrassment about acting. I notice with this company, they don't. In Denmark, even for actors, if you are to describe or feel big emotions, we always crack a joke. We undermine it, we put it down. We are good at telling the little story, but we are bad at telling the big story, for that reason. We like flat structures, like our landscape, which is like a pancake." She adds, somewhat quietly: "We like the flatness, both in our landscapes and in our social structure and in our emotional landscape as well."

We talk about her health. As was reported in the Sunday Herald last week, 2013 was an incredibly difficult year for the actress. She was diagnosed with breast cancer and underwent surgery and chemotherapy. "I am definitely better because I finished chemo. Chemo is what makes you ill."

After the treatment was over, she took, for her, the unusual decision of working away from her Copenhagen home for most of this year, first shooting a new TV series, Fortitude in London and Iceland, and then working on The James Plays in first Birmingham and then Edinburgh. The time away from home has led to a great deal of self-examination as well as examination of the cultures of the UK. She noticed the Danish influence on our language, and a sense of history. But she has already noted a difference between working on a London set, and the stage provided for her by the NTS.

"It was a huge culture shock working in England," she says as we sit on a sofa in the office of the Birmingham Rep's artistic director. "And then when I met this ensemble when we started in June, they were so different. They were mostly all Scots and just the way of being received as a stranger - the Scots do that in a very different way to the English. It's not a question of being warmer. No, it is more that: the English have so much formality and politeness compared to the Danes."

She laughs. "The Danes are considered to be quite rude you know, and I think we can come across as quite rude but we never understand it when people say that to us, because we don't intend to be. We are just so laid back that it comes across as rudeness. We are very bad at introducing people and so on, all of those structured formalities, which when I met the English people I was so overwhelmed with. The friendliness and the politeness and everyone so concerned - 'sorry to bother you' and 'we really appreciate…' - and I was really flattered. But after a while you realise: 'wait a minute, this is just a cover up, it's a front, a way of keeping you at a distance.'"

And the NTS way of working? "When I entered this company it was quite frightening because they had been working on the first two plays together every day for months. It's an amazing group of people involved, so strong and connected and I was the only new element. I was the new girl at school, except ten times that feeling because I was the cultural stranger on so many levels. Now I feel really at home with these people and embraced. But they had a very different approach … they let me be, they let me come by myself when I was ready, and that is also the Danish way. You are allowed to stand in the corner and observe a bit and then join in. That's a very different social behaviour and I felt at home in the Scottish way."

Indeed, in the play Munro has written a line for Margaret that refers to this. Margaret's early life had been hugely disrupted by her move to Scotland. "It must have been completely terrifying, to be shipped over at that age, not knowing anyone, not understanding what anyone is saying," Grabol suggests. "Every time I say the line I think 'I know that feeling.' She says: 'You taught me you can find friends anywhere, you can share food and drink and wait and see how to join in the conversation. You let me be, and you let me grow.' And that is definitely the Scottish way."

Margaret's husband, James III, is a handful. He is charming, dominant and manipulative. Margaret is sober, cool, organised. Played by Jamie Sives, James's ambitions and dreams threaten to overturn the delicate inner dynamics of both his precariously placed country and his marriage. But the two characters do, despite their history and differing temperaments, love each other.

"He's the kind of person who enters a room and changes the whole tone of the room," Grabol explains. "So living with this person who is so dominant in setting the tone, she is accustomed to adjusting to his mode, and tries to avoid the worst conflicts. He can overpower someone in a good way and a bad way.

"If, from that young an age, you have been in the company of someone that mood-changing and strong, then you might never discover your own tone, because there is something so loud being played beside you, you cannot really hear your own tone. Her story is definitely about finding her own tone and trusting that. But it takes drastic measures."

However, those measures are painful. And it is Munro's writing about the subtleties and dynamics of their relationship that Grabol, herself divorced, finds the most "beautiful".

"They genuinely love each other. Without a doubt. They are connected by fate and destiny and responsibility - their marriage was arranged when she was four - but they are also connected through love and mutual respect. It is not a story about a suppressed woman finding herself: thank God it is not that simple. It is definitely a story about very different temperaments and views on life and that is also where Rona uses the different nationalities in a very lovely way."

Grabol is a great admirer of Munro's writing. "I was amazed when I first saw the play. She writes so well. It's brilliant, it really is. It's very modern. There are no musty spider webs all over, we are not entering a dusty room. From the first page these are real people. They are human beings, and we haven't really changed, have we? All the problems are the same and the feelings are the same."

The James Plays are being performed only a month before the independence referendum. It is a topic that Grabol is well aware of, but one she cannot make dramatic pronouncement about. The character of Sarah Lund will not come out as a 'Yes' or a 'No'.

"As a foreigner I try and understand it with Danish terms: what if a part of Denmark would have their own government, I suppose it is a bit like Greenland [leaving], isn't it?" Greenland is a semi-autonomous country within Denmark.

"It is about identity really, isn't it? One can quite quickly get lost in the debate about economy and finances. Some say it would be a loss, others a gain, but it looks like something you cannot predict. So it all comes down to identity, doesn't it? I must say for me I have always felt reluctant to speak out about politics. As an actor, especially as a female actor, this whole idea that you have to be an example for something, for other women, that you have to make a statement in your art or your craft - I have always been very reluctant about that.

"To me I think that the finest duty or role of art is to mirror, hence the beautiful title of this play [The True Mirror], to mirror rather than lecture. If it was just about saying 'Yes' or 'No', it would be very poor, it would not be fulfilling its task. And I am not Scottish, and I don't have any interest in influencing people, but if you can participate in mirroring anything, whether it is cultural identity or just human identity, that is the role of the arts really."

The timing of the plays and the performances, she says, is a "gift" because of their historical resonances and the questions her particular play asks. These questions she says, are basic. Who are we? Is there a difference between us and everyone else? But the play does not offer an easy answer for audiences. As Grabol is called back to rehearsal, she adds: "I think it is kind of offensive to think that the Scottish people need an actor or a writer to tell them which way to vote. They don't. To us it is mainly a play. That is how we have to approach it - we are focused on telling a story."

The James Plays have their world premiere at the Festival Theatre as part of the Edinburgh International Festival on August 10, and run until August 22, with previews from August 5. They then transfer to the National Theatre, London, from September 10-29.