If you are of a certain age — quite a ripe one, it has to be said — you don’t even need to hear the music to be taken back at once to the pre-Beatles epoch of British DIY music. Just the name will do the trick because the name is Donegan — Peter Donegan, the 22-year-old son of the late skiffle king, Lonnie.

Put this young man’s record on and out pour the hits from his father’s heyday — Rock Island Line, Tom Dooley, Jack O’Diamonds — and countless more of the simple, ebullient numbers that lit the fuse of the rock’n’roll explosion. You can almost hear the thud of the double bass made from a broom handle, a tea chest and a length of string.

If you ever saw Lonnie Donegan perform, as I did, then meeting Peter, the eldest of three boys from his third and final marriage, is an uncanny experience — as if some top hologram artist had replicated the old man in a slightly larger format, with the inflections of face and voice perfectly intact. Members of Lonnie’s band who have played with Peter go even further and say that there have been times when they could have sworn that 40 years had been cancelled and Lonnie was standing at the microphone.

All of which gives Peter what the Americans might call an upmarket problem. On the one hand, how great it is to be able to tap in to such a source of songs as this, and to have learnt them from the mouth and the hands of the most popular skiffle player who ever lived. On the other hand, how dangerous it is to be seen as a tribute band to a man and a music whose prime was already long gone when Peter was born.

Young Donegan seems to have got the measure of this already. He plays the songs because they are exhilarating both for him and his audiences; for this reason he will probably never cut them out of his repertoire entirely. He also observes that while he misses his father sorely and feels there was so much more help and advice to come from that quarter, he wouldn’t even be doing what he is doing if Lonnie were still around, as it would have entailed a clash.

At the same time his debut album Live at the Elephant is the opening shot of an independent career. There is a strong clue to this effect in the form of a song called Just One Kiss. It has hardly ever been performed, and few people know about it since it is his own recent composition. It’s a particularly telling inclusion on the record as it is full of minor-chorded reflectiveness and is a world away from the train-crash tempo of much skiffle. If there’s more where this came from, then he will become a very different, perhaps more complex, musical proposition than his father.

Peter and the rest of the family lived in Spain from when he was 6. They were in the Costa del Sol town of Fuengirola before it was a big tourist destination. His memories are of the fishermen whom you had to speak to in Spanish as they had no English; of paella at countryside barbecues; of becoming bilingual and getting into flamenco and the distinctive rhythms that conflicted with his formal piano training.

Although Lonnie was still to tour with his old boss, the band leader Chris Barber, in the 1990s, he had reached relative obscurity, and when Peter speaks of him now he does so rather as a footballing son might speak of a brilliant father who had never benefited from the fabulous wages of the modern game.

“Dad was always Dad,” he says. “I always realised what he had done, and I was always very proud of that. I do wish he had got more recognition than he did, but obviously back in 1956 (the year Lonnie broke into the charts with Rock Island Line), the industry was completely different. He got paid something stupid; nothing, basically. But what he did get was a career that was to last more than 50 years. And if it wasn’t for my Dad there wouldn’t be the business that there is now.”

That may be a filial assessment, but it is not an extravagant one. The skiffle craze was not only massive in its own right, giving Donegan more than 20 chart entries between 1955 and 1962; it also democratised the making of popular music. Almost all of the songs were three-chord tricks, whether they came from the traditions of bluegrass, Cajun, music hall, the blues of Leadbelly or the folk tunes of Woody Guthrie. As for the rhythm section, if you had a washboard and ten thimbles from Woolworth’s, then you were a percussionist. Slung between the end of rationing and the beginning of Macmillan’s never-had-it-so-good days, skiffle (an American South term for rent-raising parties) found its moment and seized it.

“Listen to the Quarrymen,” says Peter, talking of the essentially skiffle band that was the forerunner of the Silver Beatles and hence also of the Beatles. “There’s a recording of them doing Rock Island Line at some fair, and it sounds exactly like my Dad.”

At the Albert Hall tribute concert after Lonnie’s death in 2002 Peter, who had worked with his father’s band on their last tour, found himself onstage with some legendary figures who were acknowledging their debt. “It was just amazing,” he recalls. “There was Van Morrison, Joe Cocker, Roger Daltrey, Joe Brown, Mark Knopfler. I’d never seen Joe Cocker perform, although I’d always been a great fan. So the first time I saw him was when I was playing with him.”

It’s a strange, circular situation for a young man to find himself in, knowing and being influenced by the musicians who in turn had known and been influenced by his father. The ball’s in his court and it looks as if he’s going to do some interesting things with it.

In the process he might even usher in a skiffle revival of his own. His audiences are far closer to his own age than his father’s, and they look as if they’re up for it.

The Peter Donegan Band — Live at the Elephant is released by Road Goes On Forever Records