It’s all true about Zoë Wanamaker. She’s like a wood-nymph from the Tolkien franchise. Pixie features, nut-brown eyes, a mischievous tight-lipped smile and a warm cackling laugh. Next week she makes a guest appearance at the Hampstead Arts Festival to discuss her acting career. Had she heeded the advice of her parents — Sam Wanamaker and the Canadian actress Charlotte Holland — she might never have followed them on to the stage.

‘You don’t want to go into that. It’s full of disappointment and rejection,’ they told her. ‘Do something else. Get a proper job.’ Wanamaker’s idea of a proper job was to become a painter. ‘I went to art college for a year to learn to develop my painting technique. And that was really, really [long pause] wonderful, but at the same time, a very good realisation that I didn’t want to work by myself. It’s a very lonely business trying to be a painter.’ She told her parents that she was determined to try for the stage. ‘It had been pretty apparent to them from early on. My dad always used to say, “You only want to do it to dress up.” He could have been right.’

Her parents insisted that she learn secretarial skills first and she enrolled on a shorthand course only to discover she was dyslexic. She had to cheat to acquire her diploma. She worked for months as ‘a terrible Dictaphone typist’, and then won a place at the Central School of Speech and Drama. After several years in repertory she joined the RSC and made her West End debut. Later she became a member of the National Theatre.

‘My first West End show was a play called Sylvia, about a dog. It was by E. Gurney, Gurney someone, I can’t remember his first name [A.R. Gurney], and it was a big success in New York. So they tried it in London with Maria Aitken and Robin Ellis, directed by Michael Blakemore. I was the dog. It was great. Then we closed. Sometimes the American stuff doesn’t transfer.’

Last summer she starred in Passion Play by Peter Nichols, an adultery drama. Its experimental structure places extraordinary demands on the cast. The husband and wife are each played by two actors, who represent their inner and outer selves. But the performers can’t exchange looks, or pick up cues from each other, because their presence is a metaphor. ‘It was hell to learn, that’s all I can say. Nobody ever answers each other. But it was a good piece to open up again. It’s a very honest piece of writing, frighteningly honest.’

Coming off-stage is always tricky, she says. ‘You’re buzzing a bit.’ Cocktails help her wind down. ‘A Belvedere vodka martini, straight up with a twist, very cold, very dry. I learnt that while I was doing Electra in New York. I got eczema really badly. One of the producers sent me off to her doctor, who said no yeast, no dairy, no wheat and no wine forever. You can have grain. But I don’t like whisky. I don’t like gin. So I started taking all the vodkas I could get. By the end of the run it was Belvedere that came up best.’

She goes to the theatre whenever she can. ‘I love it. I love being told stories.’ She listens to Radio 4 a lot. ‘I’m influenced by Today in Westminster. That fascinates me.’

‘How’s it all going?’

‘Badly!’

‘Why?’

‘We don’t have any heroes any more. They all seem to be so bland. We don’t have the Obamas of this world, the giants. It’s got a bit grey.’

‘Were there ever any giants?’

‘Oh God, yes. Weren’t there. Aren’t there? Weren’t we all so thrilled when Blair got in? Some of us. This was going to be our new hope. I just find everything’s a bit wet at the moment.’

She identifies three main problems. The young are disengaged from politics. The expenses scandal has destroyed trust. And politicians are underpaid.

‘How much should they get?’

‘I don’t know but it’s not enough.’

‘Sixty-six grand. It’s quite a lot.’

‘Not enough! The amount of work they have to do, and they’re not allowed to have this and that, and they do it single-handed. It’s like being an actor, to some extent, you’ve got to work all day and all night. When do they get a holiday?’

Her sympathy for politicians comes as a surprise, given that council officials were among those who tried to thwart her father’s scheme to rebuild Shakespeare’s Globe on the South Bank. The project dominated the last three decades of his life.

‘I was at drama school when the Globe bug first got Dad. But you couldn’t escape it. The plans would come out on the table. The talk was endless. We’re going to have this, have that. You had to creep away.’

Her parents moved house from Highgate to Southwark, ‘so Daddy could be near the site. Mummy cooked on a Baby Belling for two years. He suffered so many setbacks, rejection again and again. I don’t know how he did it. The tenacity of the man was extraordinary.’

Local councillors tried to shoo him off the derelict plot. ‘It was a car park. There was a court case with the council because the dustmen had to leave their dustbins there. He had to fight that. And he won.’ He faced sceptics and opponents even within the family trade.

‘All of my peers,’ says Wanamaker, ‘and all the people I was working with thought it was going to be Disneyland. It wasn’t important, they said. It was an American fantasy. It wasn’t worth it. I remember Jonathan Miller saying, “Why do we need the Globe when we’ve got Stratford?”’

‘Would it have been easier if he’d been English?’

‘Yeah. Probably. I think he always felt that, that he was slightly alien. There was a lot of resentment, a lot of antagonism towards him: he’s an American, he’s very, very rich, why doesn’t he just build it?”

In the new year the Globe will unveil its new Jacobean theatre, the Sam Wanamaker Playhouse.

‘Everything Dad had predicted about the South Bank has happened. House prices have gone up. Buildings have gone up. And just before he died it was announced that Tate Modern was going to be there, which thrilled him to bits. It’s exploded: it’s exactly what he said it should be. Like the Seine.’