To David Kane, his son was always a number: 14.14, 15/09/72, the time and date of his birth. A strange thing for a father to fixate on, perhaps, but he had little else to remember him by. Two months after September 15, 1972, his partner, Lynn, left the bedsit that they had shared in southwest London, taking the infant Matthew back to her home town of Nottingham and out of David’s life.

Then, one day shortly before Christmas 2008, David, by now, 58, opened his copy of The Times. Staring out at him from the page was a photograph of a 36-year-old man with his wife and young child. Before he read a word of the accompanying article, he was convinced that the man was his son: “I knew straight away that it was him. He’s even got his mother’s body shape, the way he was standing.”

Matt and his family were standing on the platform at St Pancras International station, having just caught the Eurostar back from Disneyland Paris — a pre-Christmas treat for Miles, 4. I had stopped them at random as part of a series of interviews I was conducting with travellers about the impact of sterling’s weakness against the euro. It was an unlikely way for David to set eyes on his son for the first time in three and a half decades.

He had been 22 when Lynn and his baby left. “She’d had enough,” he told me when we first spoke on the phone, and he didn’t mind too much at the time. While his son was growing up in the Midlands, he spent the first half of the 1970s on tour as part of the road crew for some of that decade’s more high-living bands. By the time the party ended, his links to Lynn were long gone and so, he thought, was any chance of finding his son.

Then came the fateful photo in The Times. He contacted the newspaper, confident that the Matt Kane in the photo was the one he had last seen as a two-month-old baby. My own research indicated that he was right, and I tracked down Matt’s address before writing to him to say, as gently as I could, that someone was trying to get in touch with him and that he could contact me if he wanted to know more.

It must have been a bizarre letter to receive, particularly coming from a journalist he had met only for five short minutes, but Matt sensed immediately who the “someone” must be. The next day, not long after the letter must have dropped on to his doormat, the phone rang. “That’s plausible,” he said quietly when I explained David’s story, before he began to cry a little. “There’s a photo of me and him, but I was about two months old. I can’t believe it.”

After several more emotional phone calls during the day, I left Matt with the number that David hoped he might call him on. That night, sitting in his car in front of his house, Matt contacted his father.

A little over two months later, with me there at their request to serve as a neutral third party, they met for the first time in 36 years. The designated venue was a museum café in Lincoln, near where David lives, which was almost empty when Matt and I arrived, bought tea and sat down to wait. David was not yet there, and neither of us knew what he looked like, but soon a grey-bearded man in a tired wax jacket walked past the window staring in. We knew it was him, and Matt began rubbing his hands together anxiously as he waited for his father to walk through the door. There was an awkward handshake, then a hug from Dad, before they sat down facing each other across the small table.

Neither knew anything about the other’s life, and that first conversation was a difficult one to start. Hesitantly, nervously, they began to fill in the void of the previous three and a half decades. As it turned out, the paths they had taken since 1972 could hardly have been more different.

Late that year, Lynn left with her baby because the young David, who she had first met en route to the Isle of Wight festival in 1970, was clearly not in the mood for fatherhood — “I wasn’t a very responsible type of person,” he confesses ruefully. While she brought Matt up in her father’s Nottingham home, he spent the following years as a lighting man for the Kinks, Lou Reed, Hawkwind and Captain Beefheart, touring and making a good deal of money. With an edgy laugh, he tells his son: “One great thing that happened to me around the time that you were born was LSD.”

For much of his first decade, meanwhile, Matt and his mother lived at various boarding schools, where she was a matron and he a pupil. “During the school holidays, before Mum got her own house, we would go back to Grandad’s house on a council estate,” he tells his father, always keen to stress that he had a happy childhood. While at the local comprehensive, aged 15, Matt began dating a girl called Alison. Ten years later, in 1998, she became his wife.

By then, David was also married, living in Lincoln with his wife and ten-year-old daughter, Eleanor. Earlier in the 1990s, his alcoholism had put him in hospital and he gave up the bottle, but not in exchange for a pipe and slippers. He discovered techno and “free parties”, essentially raves in fields.

The past decade has brought Matt fatherhood, with Miles born in 2004. For David, it has brought mixed fortunes. He left his wife in 1999 and since then has had a long, turbulent relationship with “the love of his life”. He spent four years living the green life in an off-grid cottage in the Lincolnshire wolds and has suffered periods of depression and a nervous breakdown. He’s happier now, sharing a friend’s home in Louth, Lincolnshire, and hard drugs are long dispensed with. David’s mother died just a month before the Lincoln meeting — which prompts Matt to say thoughtfully: “We should’ve done this a few years ago.”

Matt, like his father, is teetotal, although unlike David this is not because he has already drunk enough to last several lifetimes. His work in insolvency is going well, and Miles is in the first year of school. He leads the sort of quietly stable life that David never has: asked by his father if he is happy, he nods and replies “totally”. After the meeting draws to a close, Matt’s smile is a contented one as he says: “We are quite similar in some ways but polar opposites in most. Before, I didn’t feel like there was a hole to fill but now that it’s happened . . .”

And despite their differences, the pair do have a shared passion: 1970s rock quickly became common ground on which both could tread with ease. During their first encounter, Matt listened with apparent fascination to his father’s stories of backstage life with the bands. The latter marveled at the news that he could download bands’ back catalogues. Both seemed to find it a relief to have something to discuss other than their separate existences.

Disarmingly, however, both punctuated discussions about gigs seen and albums loved with remarkably poignant comments: “You will forgive me if I keep staring at you, won’t you?” David said at one point. Matt, meanwhile, reflected that while he is largely on the nurture rather than nature side of the debate, during teenage arguments with his mother she would sometimes say: “That was just like David.”

“Not in a bad way,” he added, eager to reassure his father that he has not been brought up to resent him. As milestones such as leaving school and getting married approached for Matt, he said, “Mum would ask if I wanted to try to find you”. Yet he appears to have inherited his dad’s ability to live with not knowing. “I never had a burning desire to do it — or not to do it,” Matt explains. Both men seemed content to leave it to fate, and agreed that the Times photo was a sign that a reunion was right.

It was the first photo that David owned of his son — he didn’t even realise that Matt’s black-and-white image of the two of them together in 1972 existed.

Four months later, when I joined them at their second meeting over a coffee in Nottingham, Matt dramatically increased his father’s family album when he handed over a set of photos charting his childhood. David, surprised, was silent for a moment before saying quietly: “A life.”

From here on, both men hope, they will share more than just photos. Matt has not yet told Miles about his grandfather. When the time is right, however, he would like his dad to meet both his wife and child. Meanwhile, he has been in contact with his half-sister Eleanor, now a university student, via Facebook, and hopes to meet her soon.

Of the two, David seems more worried about what the future will bring: “I don’t know how to be a dad,” he tells me. “I’ve nothing to offer him apart from being there. And I suspect that Matt and I have not got a lot on common, apart from being related.” That in itself is no small connection, however, and something on which to build: “We’ll have to see where it goes from here,” he tells his son. “I don’t suppose you want to come clubbing in Bradford?”