The last time I saw Mark Rylance, he was clutching an Oscar. Some of the earlier winners had been noisily triumphant. A few others were theatrically emotional. When Rylance arrived at the pressroom – trademark hat still firmly on head – he seemed sweetly bemused by the whole affair. He had the air of a man who’d gone out to buy a pint of milk and been unexpectedly handed Excalibur.

“Oh were you there?” he says in that gentle placeless burr.

"It was kind of overwhelming. There's Cate Blanchett. There's, um, that actress from Titanic . . .".

Kate Winslet?

“Yes. And they are all so professional. They work so hard. You see Cate walk up to do a presentation and her hair and make-up are all in place. So professional. But after a while, it becomes like a horse fair. It’s just a marketing device to get as many clips up there as possible.

“ The sad thing is that the public are fooled into thinking it’s a real competition. They think there really can be ‘a best’ in any craft.”

Rylance is, as you will have already gathered, very much his own creature. He is not wearing the hat today, but sartorial concerns have certainly been to the fore. Dressed in a cream three-piece suit, striking cufflinks and exquisite shirt, Rylance has made an effort for promotional duties on Steven Spielberg's adaptation of Roald Dahl's The BFG.

You would struggle to find a more conspicuously pensive fellow. Every answer is chewed over in furrowed clauses that fit together into elegant paragraphs. An original thinker who has discussed using the ancient Chinese text I Ching to make life decisions, he combines introspection with an enthusiasm for conversation.

Rylance, now 56, has been one of the UK's most admired actors for 30 years – in 1988 he delivered a famous Hamlet for the RSC – but it is only recently that he has properly embraced (or been properly embraced by) mainstream cinema. His turn as Thomas Cromwell on Wolf Hall has also made him a star on TV.

“It’s not for want of trying,” he says with a wry laugh.

Late-career surge

“I did audition a lot. One’s agent is keen to get you into film and TV because there’s more money. I was always getting myself into commitments to theatre companies. I did TV films like

The Grass Arena

, but, no, I didn’t break through like contemporaries Kenneth Branagh or Gary Oldman. You know, I had things to learn, Donald. You need to gain the confidence that you’re doing enough on film. You must resist the temptation to do too much.”

Spielberg has been at the heart of this late-career surge. Rylance is utterly charming as the titular Big Friendly Giant in The BFG. He won that Oscar – beating odds-on favourite Sylvester Stallone – for his performance as Soviet spy Rudolf Abel in the same director's Bridge of Spies. They are about to collaborate on an adaptation of Ernest Cline's cracking science fiction novel Ready Player One.

It's not an obvious partnership: Rylance, the thoughtful Shakespearean, and the billionaire who gave us Jurassic Park and Indiana Jones. But Spielberg was chasing Rylance at the very beginning of the actor's career.

"I had to turn down a part in Empire of the Sun," he confirms. "It would have paid £15,000, which was a year's earnings for me then, but I was offered a season at the National Theatre. Steven said he'd have used me a lot if things had worked out. But I don't think I was ready."

So why does this partnership work so well?

“How do you explain how you become friends with anybody?” he says. “We both love film. That’s something people maybe don’t know about me. I went to two or three films a week obsessively as a teenager. But just today I realised this thing we have in common. We both grew up in the mid-west. My family were very English. We had tea parties on the fourth of July. We were outsiders. Steven grew up Jewish in Phoenix and in northern California. Places where there were few Jewish people. So we had that.”

This is one of the great oddities in Rylance's biography. You could hardly imagine a more English presence. Indeed, in Jez Butterworth's great play Jerusalem, which played to enormous acclaim in the West End and on Broadway, Rylance inhabited a virtual embodiment of the nation. Yet he grew up almost entirely in the United States. His parents, both teachers, moved to Connecticut in 1962 and then to Wisconsin in 1969. It was at the University School of Milwaukee, where his dad taught, that he first took up acting.

Great secret

“My accent does slip,” he says. “When I arrived in England in 1978 at 18, I was shocked to find myself ‘the American’ at Rada. The English and the Americans have an intense relationship. They helped us out in the second World War. Why did it take them so f**king long? Ha ha! But I was very ignorant. I didn’t know what a pub was – American bars are not for families. I didn’t know where Manchester was. I worked hard to find an English accent.”

Rylance was, for the first decades of his career, a great secret kept by, oh, just a few hundred thousand people. Anybody who saw his turn as troubled writer John Healy in The Grass Arena would entertain the notion that this might be the nation's best actor. He followed up that Hamlet with breathlessly acclaimed turns in Romeo and Juliet and The Seagull.

In 1987, he married Claire van Kampen, theatrical polymath, and became stepfather to her two children. Seven years later, Rylance became the first artistic director of the Globe: the reconstruction of Shakespeare’s south London theatre that stages his plays in the original al fresco style. He was a great success in the role (despite forwarding the eccentric theory that there is doubt as to Shakespeare’s authorship). He must, nonetheless, occasionally regret the time the job ate up. “Um, I think I brought it on myself. I was critical of how theatres were run and fate said: ‘You have a go’,” he says. “It was an incredible opportunity and I’ll never regret it. I regret that it drew me away from my family and from my younger daughter who perished.”

Deep depression

Nataasha van Kampen, Rylance’s stepdaughter, died in 2012 after suffering a brain haemorrhage.

“We have regrets, my wife and I, about the demands the Globe placed on us and that we didn’t get home as much as we should. My inability to draw a line under how much work I should do meant I got too tired and burnt out, and I fell into deep depression. I regret that.”

I can’t imagine Mark Rylance being in any other business. There’s a gentleness to him that makes you feel protective whenever he enters the room. What would have become of him if he hadn’t made it?

There’s a long pause. Creases sink into creases.

“I don’t know whether I’ve lived,” he says. “I might have been overwhelmed with alcohol or drugs.”

Really? Has that been a temptation?

Seduced by d rugs

“Well, it’s always been around. But my work kept it in moderation. I get sensual pleasure from work. I would never go on stage high or drunk. Many actors can do that and do it very well. I have a very self-critical mind. I have to liberate myself from my self-critical mind. If I didn’t have acting to do that, I think I may have been even more seduced by drugs and alcohol to suppress that self-critical impulse.”

We meander off to discuss Keith Richards ("a very happy man, apparently"), his feelings on Ireland ("I have always dreamt of a united Ireland") and his turn opposite Cillian Murphy in Christopher Nolan's upcoming second World War epic Dunkirk ("I'm not sure Americans have heard of Dunkirk").

He seems happy to have made it this far. Whole new chapters are opening up. The secret is now known to millions. There are worse things than being still aloft in your 50s.

“Yeah. That’s right. That’s right. Many people have gone by this age. We’re doing all right. Ha ha.”