You wouldn't be so unkind as to compare life as the Doctor's companion to the business of being a Bond girl. But neither role has propelled many actors to great stardom. More than a few of those supporting players on Doctor Who used the part to build on pre-existing fame: Billie Piper, Catherine Tate, erm, Bonnie Langford. None has, however, sprung straight from the Tardis to Hollywood.

Karen Gillan might just be the exception. It helps that, since its reinvention a little less than a decade ago, Doctor Who has become a much more glamorous beast that it was when the tentacles were hosepipes and the ray-guns were washing-up liquid bottles. But Gillan also has the drive, the zing and the ambition to make it happen.

Next week we see her in a very impressive horror film – an entry to the underexploited "haunted mirror" genre – called Oculus. And at the end of July, in Marvel's puzzling superhero flick Guardians of the Galaxy, she turns up as a sadistic villain named Nebula.

Los Angeles has, for the moment, claimed the girl from Inverness.

“I suppose, technically, I do live there now,” Gillan says in a voice now flavoured with the odd Californian vowel. “But I am a nomad. I miss the weather. And all the people there of course. I miss how beautiful it is. People walk around Lincoln Heights in LA and say: ‘It’s beautiful’. Hey, you haven’t seen anything.”

Oculus was a canny project to pick for her first US film. Made on a modest budget, the picture has, in a country where horror is rarely reviewed fairly, picked up swathe of strong notices. Some sort of cult success is guaranteed. Gillan also managed to select an impressively strong role. Rather than playing a fleeing "final girl", Karen turns up as a focused obsessive who will not settle until she has proved that deadly energies lurk within an antique looking-glass.

“That’s totally right,” she says. “That’s one reason I wanted to play the character so much. We are so accustomed to a female in the lead of a horror film, but they’re usually always running away. This girl is running towards the threat. The worse it gets, the more excited she is, because it proves her theory.”

You can’t really imagine Gillan playing (to pull from suitably Scottish sources) a cow’rin, tim’rous beastie. Then again, it is easy to slip into dangerous stereotypes here. We do tend to sling unhelpful words such as “feisty” at actors with such lustrously healthy heads of red hair. That can’t be fair. We no longer expect larger actors to be jolly or blonder actors to be confused.

“Ha ha. I don’t know about that. I haven’t sold myself that way. But maybe there is a common theme of ‘feisty’ to my roles: feisty outside and vulnerable within. There’s something fun about red hair flying around in adventure situations. Isn’t there?”

We go on to agree that, up in the northwestern corner of Europe, from where we both hail, red hair is really not worth remarking upon. There’s thousands of us on the streets of Belfast and Inverness.

Gillan was born in that remote city some 26 years ago. Her father was a fine singer and she began tinkling the piano with him when she was still a tiny girl. She began acting in local theatre groups and, when she was 16, moved to Edinburgh. A year later she was seeking work in London. It sounds as if she was a brave teen.

“Well, I went down there because I was at college. Then I dropped out after just two months. I was working in a pub. That was scary. What am I going to do? What happens next?”

What did happen next was that she picked up some modelling work. We hear such horror stories about this business. The very notion of the teenage Gillan going among white slavers fills one with dread.

“I heard many a story also,” she says. “It is maybe risky to have 25-year-old Russian girls in the business. I moved away at 17, but I was still in my own country. It was all very normal, I have to say. I didn’t meet anyone with an eating disorder.”

Anyway, as she explains, modelling never threatened to get in the way of acting. She was forever frustrating her agency by vanishing to auditions. Eventually, the Doctor Who gig lumbered over the horizon. As she tells it, there were no enormous negotiations, no endless call-backs, no tedious extended readings.

"It was a total rush. I went in for the last audition with Matt Smith. They hugged me and I knew then that I'd got it. It's never happened like that before, I can tell you."

Among the tweaks that helped the new Doctor Who become a phenomenon for the age was the decision to flesh out the Time Lord's companion. Billie Piper and others had already prepared the ground, but Amelia "Amy" Pond, the character played by Gillan, has a particularly rich hinterland. She gets married. She has a child. There really was a great deal going on there.

“Oh yes, that’s what I loved so much about her. We didn’t just learn about her time with the doctor. We met her as a child. We saw her get married. She had a kid. We experienced all of that with her. And they showed sides of the character that wasn’t just running around with the Doctor. Other characters from her life were introduced. She was a real character.”

Her eyes and ears are our eyes and ears.

"That's right. Doctor Who really is the story of the companions. We learn about him through them."

It must have been tricky for Gillan to cope with that sudden avalanche of fame. She was, after all, just into her 20s when Amy made her first appearance. Some survey I've just made up tells us that 62 per cent of internet traffic is taken up with conversations on the minutia of Doctor Who. One day she was a struggling actor; the next she was a well-worn hashtag. That's an education and a half.

“Well, I learned quickly that you don’t Google yourself,” she says. “We were being slated before we even shot anything. That is pretty tricky. There was a big transition. Suddenly everything we did was analysed. Then they turned to our personal lives. Yes, it really is like going from nought to hundred overnight. How did that happen?”

All reports confirm Gillan's story that she and Steven Moffat, the show's supreme boss, mutually agreed to manoeuvre Amy out of the series in late 2012. But those two years beside the Doctor have served her well. Just look at the publicity that gathered around her when, at Comic-Con last year, she revealed the bald head she was required to sport for Guardians of the Galaxy.

Of course, as we mentioned earlier, she does (or did) have famous hair.

“Famous hair? That’s a new one. It was a really cool thing to do. Now, I can look back on my 20s and think: I did that. I shaved my hair. I still have the hair in a wig. It’s sitting on a mannequin head in my bedroom. So I see it every time I wake up.”

Did people treat her differently when she was bald?

“Definitely. When I was completely bald people were a little intimidated by me. Or sympathetic. That was strange.”

Some sort of space adventure involving a killer racoon, Guardians of the Galaxy is sure to scare up Gillan more chatter. Since finishing that picture, she has shot a pilot for the sitcom Selfie, a contemporary retelling of Pygmalion featuring the pride of Inverness as a solipsistic valley girl. "I'm happy being American because I used to play with my Barbies in an American accent."

Then there’s a “really exciting” project that, though apparently close to confirmation, she’s not allowed to talk about.

What a busy, busy time Gillan has had over the past five years. Doctor Who is receding in the rear-view mirror, but she still feels very close to that breakthrough role.

“We went to conventions and people told me it saved their lives. And I’ve met so many baby Amelias. They bring them along to meet me – these tiny babies named after her.”

Aww! Now, that’s what I call a legacy.