'We weren't brainwashed': Jemma Redgrave on Doctor Who, family tragedy and growing up in a radical acting clan

The last time Jemma Redgrave gave a proper, full-on newspaper interview was in July 2010, just a couple of months after the deaths of her aunt Lynn Redgrave and her father Corin Redgrave and just over a year after the skiing accident that killed her cousin, Natasha Richardson – an awful succession of loss that the interviewer described as giving her face "the look that grief gives, as if a layer has been washed away".

Three years later, and Redgrave appears outwardly restored – friendly, warm and unpretentious, with an unexpectedly hearty laugh that wouldn't disgrace Basil Brush. If she remains huddled under her coat in the well-heated bowels of the Soho Hotel in London, then it's because today she is sniffing her way through a cold.

"It was a couple of months after he [Corin] died, so I was quite raw," she says of that 2010 interview. "I still feel the same now, just not with the same intensity." We talk more about her father and other relatives later, and not altogether mellifluously when I reveal that some of my research came by way of a biography of the Redgraves despised by her family.

First and more happily, however, we discuss her work. Since leaving drama school, Redgrave has been a regular on television, most prominently as the titular Victorian doctor in ITV's Bramwell. Thanks to its huge global fanbase, however, her role in Doctor Who, in which she debuted last year as Kate Stewart (the daughter of the much-loved Brigadier Lethbridge-Stewart, played by the late Nicholas Courtney from 1968 to 1989) is set to eclipse all that has gone before, when she returns in the 50th- anniversary episode "The Day of the Doctor".

So far we know that this "love letter to the fans" has been filmed in both 2D and 3D, and will see the return of David Tennant and Billie Piper alongside Matt Smith and Jenna Coleman, as well as John Hurt as a previously unknown incarnation of the Doctor, plus Daleks, Zygons and a visit to Elizabethan England. Otherwise, a strict omerta prevails around the 75-minute episode that will be shown simultaneously around the world as well as in cinemas. That's next Saturday – quite the event.

"What can I tell you about the 50th anniversary? Practically nothing," she says, giving me a first taste of her pleasingly full-throttle laugh. "When the job offer came in my agent said, 'You mustn't tell anybody about this,' and I thought, 'What am I going to tell the kids?' It's like joining M15." The cat finally exited the bag when scenes were filmed in Trafalgar Square. "The news hit the Twittersphere and within half-an-hour of our being there, there were people with Tom Baker scarves on… people with Tardis safety covers on their iPhones," she says. "It was a huge relief to be able to tell people."

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Redgrave's peak Doctor Who-viewing years were the early 1970s, when, classically, she'd watch from behind the sofa. "I would then have terrible nightmares," she says. "My dad said he would take me to the BBC studios so I could see the Daleks – and that frightened me even more." Does she meet one in the anniversary special? "Can I tell?" she asks the publicist sitting in on the interview, who signals her assent. "In that case, yes, I come across a Dalek. There was no acting required. It was a scarifying moment."

Anything else she can tell? "I work with more than one Doctor… oh, and I worked with more than one Tardis as well." Intriguing, or at least it will be to Whovians. "The community of Who fans have been very kind to me," she says. "Brigadier Lethbridge-Stewart was such a loved character and I think people were very open to his daughter making an appearance and, hopefully, touch wood, making more appearances in the future." So, she'll be back? "I think Peter Capaldi is a very exciting prospect as the new Doctor, so that would be wonderful."

Born in January 1965, Redgrave is five days younger than her cousin Joely Richardson, whose parents are Vanessa Redgrave and the film and play director Tony Richardson; Joely's sister, Natasha Richardson, was born two years earlier. In the flesh, she bears a far more striking resemblance to her late cousin than she does when photographed – or, at least, I'd never noticed such similarity before.

Her paternal grandparents were the actors Sir Michael Redgrave and Rachel Kempson, an acting dynasty, if you like… although Aunt Vanessa doesn't like, insisting that "dynasty implies power – we're a family of professional actors. It's like coming from a family of carpenters or plumbers." "I think that's about right," agrees Jemma (née Jemima). "I associate dynasties with huge corporations… the Murdochs… it feels like a family and quite a few of us are actors."

When did she first become aware that she belonged to this extraordinary clan? "I remember one of my teachers at primary school used to call me Vanessa by mistake, and I couldn't understand why and then, of course, later it became clear," she says. "It just seemed very normal to me – like everybody's family seems normal until you realise no one else's family is like that." Was it inevitable that she would follow in the family profession? "No, not at all. None of my brothers are actors – I've got three brothers – Luke is a cameraman, Harvey is a civil servant and Arden is training to be a primary-school teacher. A mixed bag.

"I remember once on my grandmother's birthday, my dad was filming In the Name of the Father (the 1993 Daniel Day-Lewis film about the Guildford Four) in Ireland and my aunt and my brothers… a big lot of family… were driving round from here to there in a minibus, having a lovely time and breaking into songs, and my brother Luke heard Harvey mutter to himself, 'I was born into the wrong family.'"

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Her own sons with barrister husband Tim Owen, Gabriel and Alfie, are aged 19 and 13; Gabriel has just started an English degree at Sussex University. Are there any signs of a new generation of thespians? "There are going to be one or two more… possibly… but I think it's important that they speak for themselves," she says – a statement in stark contrast to Laurence's Olivier's very public announcement of the birth of Vanessa Redgrave, after a performance of Hamlet at the Old Vic, that "Laertes [played by Michael Redgrave] has a daughter." Vanessa could hardly grow up to be an accountant after that.

It was Jemma Redgrave's grandmother, Rachel Kempson, who took her – aged five – to her first play, Peter Brook's RSC production of A Midsummer Night's Dream, followed by more Shakespeare, watching her father in Julius Caesar and Antony and Cleopatra. "Complicated theatre really… not Chitty Chitty Bang Bang, which I took my children to see."

Her parents, Corin and former fashion model Deidre Hamilton-Hill, divorced when Redgrave was nine, by which time her father, like her aunt Vanessa, was deeply involved in far-left politics in the shape of the Workers Revolutionary Party (WRP). "We'd been taken to demonstrations when I was very young," she recalls. "This was the late 1960s, early 1970s, and everybody was demonstrating about something.

"It's difficult to explain it now… you know the whole Ed Miliband thing with the Daily Mail and 'it's very important to know where he comes from… very, very left-wing views were expressed round his breakfast table'… well, they were discussed round the breakfast tables of a lot of people who grew up at that time. The children of those people weren't brainwashed."

Certainly this child isn't without her own political causes: Redgrave was a prominent member of the Stop the War movement protesting at Blair and Bush's 2003 invasion of Iraq. ("There's no joy in being proved right.") She also helps at a Red Cross refugee centre in Islington, north London. What she can't stand is that any political movement would espouse a cause to the detriment of family life. "I resented the WRP, because my dad was unavailable to me and to my brother because there were such extreme demands made on everybody who became a member of that party," she says.

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In her book, To Be a Redgrave, her mother recalls Jemma and her cousins Natasha and Joely sitting round the kitchen table discussing how much they hated the WRP. "Vanessa was involved in the WRP for a while so we did have a similar experience, yeah," she says. I add that I'm surprised that she has stated that she has never read her mother's autobiography. "My mum was very angry with my dad for a very long time and I didn't really want to divide my loyalty," she explains. "The least complicated path through that particular difficulty was not to read it."

Another contentious book is one that I had blithely borrowed from my local library, The House of Redgrave by Tim Adler, unaware that it had been lambasted by the family for an outrageous false claim that Vanessa Redgrave had once come home to find her husband, Tony Richardson (director of the original Royal Court production of Look Back in Anger and Oscar-winning British New Wave film-maker), in bed with her father, Sir Michael Redgrave.

"You can't libel the dead so [Adler] can make up what he likes… I don't even want to comment on it," she says, before adding: "That book was written by a man who got in touch with my cousin Tasha and said that he wanted to write a book about her dad and that he was a huge fan. She did a bit of investigating and she said that she wasn't going to help him. He wasn't a huge fan of Tony's. This is a man whose work was groundbreaking and changed the landscape of theatrical and cinematic culture in this country. And to reduce him to his sexuality… it's… yeah."

A long silence follows and we talk about other things to get the conversation flowing again – of her recent house move across north London, her cottage in Wales and her pet Labrador. And then our time is up and she is whisked off for a photo-session with the marvellous Dan, who soon has her booming with laughter again. After the shoot, I tell her that I will return the despised book to the library forthwith. "Or burn it," she says. "No we can't start burning books. Oh, all right – perhaps just this once."

'The Day of the Doctor', the 50th-anniversary special, is on BBC1 and in cinemas on Saturday

UPDATE: Since this interview was published, we have heard from Tim Adler who tells us that Natasha Richardson did not say she was not going to help him write her father’s biography and, in her last email to Mr Adler, dated 3 March 2009 - 16 days before her death - Natasha agreed in principle to this idea and asked her literary agent to draw up an agreement. He adds that it is not right to suggest that he misrepresented himself as a fan of Tony Richardson’s work or was somehow underhand in his dealings with the family. He points out that the Independent’s reviewer thought too much of the book was devoted to Tony at the expense of other members of the Redgrave family.