And you think getting a doctor's appointment is difficult – try getting one with the Doctor's assistant, flame-haired Scottish actress Karen Gillan, aka Tardis companion Amy Pond. Nine solid months of 11-day fortnights filming Doctor Who in Cardiff, on-call from six in the morning until way past dusk, means that clearing a space in Gillan's diary can seem harder than time travel itself. But now here she is, heeled-up and tottering carefully out of a taxi on to London's Portobello Road.

Matt Smith has playfully likened his co-star to a praying mantis, but with her curtains of red hair and long legs fully displayed below her trademark micro-mini and extended further by the high-heeled boots, it's a flamingo that springs to my mind. "Going somewhere for the weekend?" I ask, helping her with her suitcase. I imagine the overnight sleeper to visit her parents in Inverness, or the shorter trip to see her boyfriend in Ipswich. "New York to promote Doctor Who," she replies in her Highlands-Scottish accent. "Then to LA to do The Late Late Show."

It's not the 23-year-old actress's first promo visit to New York – last year she was shocked to see fans queuing round the block, some of them dressed as her. "I think it's the biggest show on BBC America now," she says, which helps explain why the first episode of the new series was filmed in Utah's Monument Valley – the backdrop to so many John Ford westerns – and why Smith and Gillan are being whisked off Stateside instead of appearing on Graham Norton (Norton had to do with ex-Doctor Who stars David Tennant and Catherine Tate). "We didn't think anyone knew what it was over there."

Certainly no one knew who Gillan was when she was cast at the same time as the only slightly-less-well-known Matt Smith as the 11th Doctor, but her Amy Pond quickly won over doubters with her dry, sarcastic wit and short skirts. In fact, let's be honest about this, large sections of the British public fell hopelessly in love with her. She was "the sexiest Doctor's assistant ever" when she first turned up in a kissogram police uniform – in fact too sexy for some of the more prurient sections of the media. "I think it's really funny," says Gillan with a laugh (she laughs a lot – in a girlish but unforced sort of way). "It's not too sexy... it's really not. It's all part of the history of the show that the assistants are a bit sexy – I thought so anyway."

Indeed it is. Lest we forget, there was Katy Manning (Jo Grant in the Jon Pertwee years), who infamously posed naked with a Dalek, or Nicola Bryant, Peri to Peter Davidson's Doctor, who was introduced specifically to sex up the show. But you can probably chart the demise of the paternalistic Reithian tradition at the BBC with Doctor Who's changing relationship with his young female assistants.

It was once fatherly, or even grandfatherly (I'm going right back to William Hartnell here), but the re-booted 21st-century version has seen the assistants becoming romantically entangled with their various Doctors – Billie Piper's Rose Tyler with Christopher Ecclestone and Freema Agyeman's Martha, unrequitedly with David Tennant. But neither companions quite so brazenly attempted to seduce their Doctor as did Gillan's Amy Pond last season, throwing herself on her bed and suggesting that the Time Lord "sort her out" to calm her pre-wedding jitters. Mediawatch-UK, the current form of Mary Whitehouse's censorship movement, accused the show of bringing casual sex to Saturday tea-time. "I read the complaints," she says. "But the Doctor isn't reciprocating, so it's not inappropriate. We're not sending out bad messages."

What is without doubt is that Gillan and Smith have excellent screen chemistry, rather like Diana Rigg and Patrick Macnee in the heyday of The Avengers. There's a lot of smart banter, while the writers play on the suggestion that it's the Doctor that Amy really loves and not her hapless husband, Rory (played by Arthur Darvill). "I don't see it as a will they/won't they relationship, in all honesty," says Gillan. "At heart they're just best mates – almost verging on brother and sister. Rory's the one for her... I think. Having a married couple on the Tardis helps to define properly the relationship between the Doctor and Amy ... Having said that, their relationship is about to get weirder."

How so, she's not telling, or whether Amy really is pregnant (this season's teaser) – because Doctor Who storylines are surrounded with more secrecy than a Premiership footballer's extra-marital dalliances. Gillan's audition for the role involved a clandestine visit to BBC Television Centre, where she had to give the receptionist the password "panic moon" ("it's an anagram of companion") and have her private life vetted; she didn't tell her mother that she'd won the role because her mum is a "proper diehard Whovian. She's got a Tardis money-bag, and Dalek bubble-bath" and might have blabbed.

Gillan was born in Inverness in 1987, when Bonnie Langford was in the Tardis and definitely not getting it on with Sylvester McCoy. Her mother, Marie, is from Glasgow and her dad, John, who manages a care home, is from Sunderland; he is a bit of a performer himself it seems, at the local pub's open-mic nights. Their only child Karen was, by her own account, "really, really shy. It was quite difficult for me then to get into performing because I was so nervous".

She was a determined wee lass, however, and despite her father teaching her to read, write and play chess before she even started school, she never really got on with that part of her education that didn't involve drama. Her main memory of school is of being teased for being a redhead. "I don't know why, though, there are loads of us in Scotland. It's almost a majority," she laughs. "I was teased but it's OK, because I used to like my hair even at that age." In fact, Amy Pond has become something of a role model for young redheads. "I get letters from them – it makes me really happy."

She did dye half her hair pink at one stage, in honour of a Beyoncé video, as well as getting her belly-button pierced, but that is as far her rebellious stage went, because by the time she was 16, she had moved to Edinburgh to attend drama school, and a year later had enrolled at Bonnie Langford's alma mater, the Italia Conti stage school in London. She left after a few months, however, to take a job in the Scottish detective drama Rebus, but the filming only lasted eight weeks. Wasn't that a bit rash? "No, no. It's good to go with your gut instincts in life," she says. "You just should. Even if it doesn't work out, something good will come out of it. Having said that, after I did that, I then got a job in a pub in London..."

The Pilgrim in Kennington will perhaps one day live up to its name and become a destination for Whovian tour groups, Gillan having worked behind the bar there for a year while living "in this flat with two really stinky boys... I did love it, though".

Rather more lucrative was the modelling work that came in after she was scouted. "I was tending to do quite a lot of hair stuff actually. I don't like modelling, but in terms of the places I got to visit, it was great." In the meantime, she was picking up bits of TV work, most consistently on Channel 4's The Kevin Bishop Show, during which she got to impersonate Katy Perry and Angelina Jolie. Can she do her Jolie impersonation? "I'll do this," she says, sticking her lips out into a trout-mouth. "I wouldn't really call it an impersonation... I mean, it was nothing like her." But she enjoyed, she says, because "it was a kind of comedy crash course".

But it's the modelling background rather than the comedic one that will come in handy for her latest role, playing iconic Sixties model Jean Shrimpton, in a BBC4 biopic about The Shrimp's relationship with the photographer David Bailey, who has yet to be cast. "Jean Shrimpton. Yes, who also caused a bit of a stir with a mini skirt," says Gillan, fascinated when I mention that I caught sight of Shrimpton recently on an episode of Channel 4's Country House Rescue, running a hotel down in Cornwall. "Thanks for that. I'm going to watch that definitely, because footage of her is so hard to find.

"She's got quite a soft and breathy, well-spoken English, so that's what I'm practising at the moment, and she's quite enigmatic, so that's something that's going to have to be acted on my part! I read her autobiography and I'm also watching films from that period just to get a feel for the way they talked. The other night I watched Blow-Up, which is based on David Bailey."

By coincidence, not only is Gillan an ex-model who wears miniskirts, but her boyfriend, Patrick Green, is a photographer. "Yes, there are quite a few similarities, actually. The fact that she didn't really like modelling and I didn't really like it as well – it's something I don't have to work to understand."

The best that the paparazzi have managed so far is to snap Karen and Patrick buying a bonsai tree in a Suffolk garden centre, and she adamantly won't talk about her boyfriend, suffice to say that they've been going out for six years, he lives in Ipswich, likes cars and that he owns a tortoise called Brian. Oh, and his mum opens her fan mail. "She tells me that there's been nothing scary but I'm not sure. Generally it's from young girls and... some older men. There's one guy who sends a postcard from every single place he goes to and he travels quite consistently so I've got this update of his travels, which is really nice..."

At which point I exchange raised eyebrows with Gillan's PR, who has just come to her rescue with a can of Coke and a packet of Hula Hoops ("I feel quite dizzy... my fruit salad failed me"). Her innocence in such matters as obsessive fans will eventually wear off, no doubt, but hopefully not all her unaffectedness. For the moment she is happy, carefree and refreshingly ordinary – a tribute to her upbringing. And although she's vaguely aware that she's now part of a national institution, fame itself is something that impinges only rarely, so busy is she producing the thing that is making her celebrated in the first place.

"I have no sense of being famous – you're just working. And then you'll have a random day in London when you'll do some press and it creeps into your awareness that this goes out – that what you do every day goes out to televisions right across the country. I just forget. Then, it's a bit like being a pop star."

Life in Cardiff sounds almost monastic, even with the odd jaunt such as a recent one to see Arcade Fire in concert (Smith was mobbed apparently, but Gillan left to dance unmolested). "Filming the show becomes your life in a way and I really worry that I don't have anything else to talk about," she says. "Everything's Doctor Who. We don't have a social life or anything, because you just can't, and film the next day.

"I watch stuff on YouTube every night – that's what I do in the hour that we have between filming and looking forward to our houmous and Nairn's oatcakes. That's what Matt and I have in the evenings... it's depressing," she laughs. "I've been watching loads of stuff about cults at the moment – about Scientology, which I find fascinating." How fascinating? Will we see her standing next to Tom Cruise one day – towering over the Ron L Hubbard cheerleader-in-chief? "Never... never. Absolutely not. But that doesn't stop it being interesting."

In the meantime, Amy Pond is set to become more complex, it seems. "Her life was messed up by this guy who dropped out of the sky and she thought she was crazy all her life and then the guy comes back... You're not going to be normal. So she's a bit of a broken person – that's why she's got a feisty exterior, because people tend to put walls up. But in this next series we're going to watch those walls come down and we're going to see what a vulnerable person she really is."

And will the hemlines come down along with the walls? "Now she's more settled in herself we'll see that reflected in her outfits. I didn't want to get rid of the short skirts completely – because that's what young girls wear. I know that because I am one of them."