It’s not a career path any Danish actor would dare to imagine. Even for the most celebrated of that country’s stars, the prospects of working abroad remain slim. So although Sofie Gråbøl was one of Denmark’s best-known actors long before being cast as straight-talking detective Sarah Lund in The Killing, she had little reason to ponder a life far beyond her native Copenhagen.

Why would she? There, she could be with her two children, now aged nine and twelve, take roles in the Royal Danish Playhouse, work with Lars von Trier on The Boss of it All and pick up TV work just as she had done since the age of 17 by landing a part alongside Donald Sutherland and Max von Sydow in Oviri. ‘I’ve always been very privileged in Denmark that I’ve been able to move freely between film and theatre and television series,’ she says. ‘It was never an ambition of mine to go outside of Denmark to work. Since I’ve had children, I haven’t said yes to anything abroad because obviously I need to be here with them.’

Even when The Killing won a devoted following in the UK, cultivating a taste for subtitled telly, dreich Nordic cityscapes and big woolly jumpers, it was by no means certain that she’d capitalise on the international interest. After the third and final series came to an end in 2012, she went straight into period dress for a stage version of Ingmar Bergman’s Fanny and Alexander in Copenhagen.

And then came the bad news when she was diagnosed with breast cancer. ‘I was ill all of last year,’ says the 45-year-old today. ‘Not until this year has working away from home really fitted into my life. Also my kids are at an age now where it’s possible to go away for a while.’ She’s too smart to resort to glib generalisations about how going through chemotherapy and surgery changed her, but change her it did. And she accepts it may have played a part in her decision to up sticks for half a year and join the National Theatre of Scotland to play the Danish Queen Margaret in the final piece of Rona Munro’s trilogy of The James Plays. That’s in addition to filming in Iceland with Tam Dean Burn, Michael Gambon and Christopher Eccleston in Fortitude, a 12-part TV series written by Edinburgh’s Simon Donald in which she plays the governor of an Arctic town.

‘It definitely changed the perspective radically, but I haven’t really formulated what it has changed,’ she says. ‘I could say that it makes me appreciate life more but it also does the opposite. It’s so complex that it gets banal if you try to squeeze it into a sentence. But maybe it is one of the reasons why I took this role. If I’d been offered it a few years back, maybe I would have wanted it just as much but the fear of doing it might have overshadowed the desire to do it. I remember saying to somebody, “I can’t lie on my death bed (whenever that is) not having done this just because I was frightened”. So, yes, I guess it has changed something.’

The inaugural production of NTS artistic director Laurie Sansom and the centrepiece of the Edinburgh International Festival’s drama programme, the three plays concern a little-known period of Scottish history. In the first, Munro tells the story of James I who became king at the age of 12 in 1406, but was held by the English for 18 years before he could do anything about it. The second is about James II who inherited the crown as a boy and had to wait until adulthood before he could take control of Scotland’s feuding families. The series concludes with James III who married Queen Margaret, the power behind the throne, in 1469 at a time of political instability.

If you’re feeling embarrassed about your ignorance of medieval history, you’re not alone: Gråbøl says Queen Margaret is ‘completely unknown’ in Denmark too. What grabbed her instead was the richness of the writing. ‘It has this substance of historical truth, but at the same time, it’s not dusty. Sometimes when we tell stories that take place in historic times, you feel like you’re in a history class, but Rona goes the opposite way with it. She lets the period inspire her and then she plays with it. It’s really modern, funny and clever.’

In conversation, Gråbøl is sharp, open and engaged, talking with passion and intelligence without taking herself too seriously. Although she speaks perfect English, she says it’s still tricky to convey her sense of humour (‘it’s hard to do jokes in other languages’). In other words, Gråbøl is the opposite of Sarah Lund, the unemotional loner she regarded as a female Clint Eastwood. The dryness of that character came as a surprise to Danish viewers and is not typical of her work. So what should we expect from her as Queen Margaret? Will she be wearing a woolly jumper?

‘No!’ she laughs. ‘It’s very different. Even though Sarah Lund is a character that means a lot to me, it’s only one of many characters I’ve played. That’s a benefit of coming to another country: if you were a piece of paper, there’s not a lot written on you. In my mind, I don’t think people expect anything.’

Those of us addicted to the Scandi-dramas that followed in The Killing’s wake – from the political machinations of Borgen to the cross-border policing of The Bridge – are used to spotting familiar faces popping up as different characters. ‘Has Denmark run out of TV actors?’ fretted one newspaper blog, providing a Venn diagram to illustrate the looming national crisis. If the threat was exaggerated, Gråbøl is nonetheless thrilled to find herself an unknown quantity in a foreign country with the chance to start afresh.

‘I’m 45 now and I started acting when I was 17. I suddenly started getting that feeling that I had in my very first years; the feeling of entering something completely unknown to me. In Denmark, with every project you work on you meet someone you’ve worked with before or you know what they’ve done. I’m sure this is the same in Scotland. You always have this history you carry with you. For me to go somewhere and not know anyone and for them not to know me … I feel like everything is opening up in a way I haven’t experienced since I was very young.’

She is also looking forward to getting under the skin of Scottish culture, especially as the country heads towards September’s independence referendum. ‘One of the powerful things about exchanging stories is that on the one hand, you’re just entertained and on the other hand, you really get to learn about different cultures and their way of looking at life. That was my experience with The Killing: it was just a whodunit, a classical crime story that caught on, but very quickly viewers in other countries started being interested in the language, what the furniture looked like and the political system. In Denmark we have always looked out to the world, but it’s interesting when it goes the other way and I will definitely start looking more into Scottish culture now. Until you start expressing who you are, no one else can know who you are.’

The James Plays, Festival Theatre, Nicolson Street, 0131 473 2000, 10–22 Aug (not 11), times vary, £15–£35.

The James Plays - Sofie Gråbøl Interview