When I first met Daniel Radcliffe to talk about a doing a South Bank Show with him, it was in Jude Law's dressing room, after Jude had given his stunning warrior performance as Henry V. Daniel looked trashed out (night filming), excessively nervous in that martial company of master performers, and shy, as if he felt he did not belong in such a world of heavy theatre.

Illustration by David Foldvari.

The next time I met him in a dressing room was some months on in his own in New York. He had just come off stage after his performance in The Cripple of Inishmaan. The audience had given him a standing ovation. The reviews were to be magnificent. He was still nervous but no longer an outsider. In one bound, it seemed he had freed himself from a lifetime on celluloid. If he had felt lost before in the theatre he'd certainly found himself on stage as Cripple Billy. He is a serious contender on stage now, already one of the best of his generation.

How has he done it? In the SBS I wanted to look at that remarkable leap. Without Rada, without rep, without university or the new drama schools such as Eton. With just a couple of stage ventures behind him: his bold, even reckless, performance in Equus and his admired accomplishment as a song-and-dance man on Broadway in How to Succeed in Business Without Really Trying.

But this is completely different. In Michael Grandage's faultless production and interpretation of Inishmaan, which transferred across the Atlantic without any compromises along the way, he shines out across the proscenium arch as brightly and commandingly as ever he did as the young wizard of Hogwarts.

When he got that role at the age of 11, "I cried", he said. And also at the first big press conference he said: "I have read the least Harry Potter of anybody in my class." That kind of self-deprecation and honesty has never left him. He worked with Gary Oldman in the films and told me how he admired Oldman and David Thewlis and, in Equus, Richard Griffiths, and how he'd bombarded them for advice. He then said he even cross-questioned Oldman about how he should behave on his first date!

On Harry Potter, he had singing lessons and action lessons but not one drama lesson. He told me: "I could have done with one, especially when I look at those first films." His honesty has no barriers; briefly we referred to his late teenage drinking jag that stopped four years ago when he was 20. Compared with the self-destruction of many of his contemporary celebrities, it was fairly normal wild oats stuff for a while. But a price he pays is that in several newspapers every small reference to it is inflated as if it was last week, to remind us all of that brief time.

He has an extreme politeness that is almost Biggin Hill in its Englishness. "Can I say shitty on your programme?" And after a deep breath he said: "My favourite John Water's quote is, if you go to someone's house and they don't have books, don't fuck them."

He soaks up information almost rabidly. When he goes into a bookshop, he always buys the books he thinks he might never come across again. We did some filming next to Brooklyn Bridge, in which he's interested, and as we strolled along the Waterside I enjoyed an enthusiastic cascade about the building of that great connection across the East River.

John Lennon once said that the Beatles were more famous than Jesus Christ. He was hammered for saying that, but he was right. Popular culture now carries all before it. For a generation and, if we look at the way the Harry Potter films and books are constantly finding new young audiences, for generations to come, Daniel was in danger of being famously imprisoned in that part.

On our set, this was replicated in a small way when he came to do the interview in Sardi's (the Broadway theatre restaurant, floor-to-ceiling caricatures of famous actors, Radcliffe recently put on the wall) and there was a circle of people looking after him. Security, make-up, press. His kind of celebrity means he's a target for the type of enthusiasm that can ruin your day or worse – back to Lennon.

In New York, in the West Village, he's making a determined attempt to lead a normal life. He reminded me of the way Paul McCartney steered his life through global and instant recognition to determined ordinariness. He walks along the river. He watches acres of television, especially drama and films. A year or so ago, he discovered that he had no work to do (for two or three weeks!) and that he had never had a hobby. So he took up rock climbing – the sort you do inside, harnessed and up a wall, but still it looked tough enough when we filmed him doing it.

I was interested that when he talked about filming Harry Potter the first people he referred to were the crew. That, he said, was the best thing, getting to know the crew. Each person in our small crew was greeted and questions were asked about equipment. I wouldn't be surprised if, like Ron Howard, who was a famous boy star in America, he doesn't head for film direction sooner or later.

Certainly, in the first post-Potter film he did, The Woman in Black, he was not only exceptionally good and helped make that film an unexpected roaring success. He was also very keen when we talked about it to point out in detail how the director had arrived at particular shots and angles.One way that he warms up for a part is by listening to music. "I know the song I want to play if I want to get angry… I create a playlist for every part." The main song he plays for Cripple Billy is Wasting My Young Years by London Grammar.

Like many stars, he works harder than anyone around him. With Janis Price, who trained him in the details of how he should move as someone who had a particular form of cerebral palsy, he reminded me of young ballet dancers going through the pain barrier time and again as they will not let go until it is muscle-perfect.

He has a gift – Richard Burton, also "untrained", had it in abundance – of sitting still and silent and yet the centre of attention. He has complete believability.

He's a bold young man. The 17-year-old Cripple Billy is about as far away from the 17-year-old Harry Potter as could be and yet Radcliffe blazes across the stalls in the theatre every bit as much as he does in the cinema.

He has released himself from that profoundly cosseted cage of rare celebrity. When we came out of his dressing room and out of the theatre in New York, having played to the Broadway crowd who loved his performance, he was met with a street-jammed and screaming crowd of young people who wanted Harry Potter. He smiled and signed obligingly but apart from these big occasions when he is known to be available to his young public, he says, rather proudly: "When people talk to me in the street now they usually call me Daniel rather than Harry Potter."

He is totally open about the fact that he was nervous as hell after the Harry Potters came to an end. Many doubtless well-meaning friends told him that his career was finished and the best was behind him. "What did you say?" "No, it's not!" He reached back to the age of 10 when he played David Copperfield in a BBC adaptation and first worked alongside actors and started to try to act. "And I loved it! I loved it more than anyone I have ever seen love it." He still does.

Daniel Radcliffe is the subject of The South Bank Show on Sky Arts on 26 June