Leo Green was 16 when he and the rest of his family saw Frank Sinatra perform at London’s Royal Albert Hall in 1989.

For the music-mad teenager — son of the jazz saxophonist, broadcaster and writer Benny Green — the concert was a magical experience. But even that paled in comparison to what followed.

He could barely believe it when his father said: “Let’s go backstage and say hello,” and was in dreamland when Sinatra spoke to him and his siblings.

“Hey kids, how are you? Hope you enjoyed the show and you’re not giving your folks a hard time.”

Thirty years on, Green looks back in wonderment: “I still have to ask my brothers if that actually happened.”

And he retains that almost childlike sense of awe, despite a career as a musician, musical director, radio presenter and promoter in which he has worked with a litany of rock royalty.

Among a list too lengthy to recount in full, he has played sax with Sir Paul McCartney, Bob Dylan, Eric Clapton, Ronnie Wood, Jeff Beck, Little Richard, Chuck Berry, Bo Diddley, Bryan Ferry and Noel Gallagher.

Green has also toured extensively with the likes of Jerry Lee Lewis, Van Morrison and Jools Holland, has his own orchestra, and since 2011, has been director of BluesFest, the annual concerts, which down the years, have featured an eclectic mix of stellar names. “Everything comes from the blues,” he reasons.

Unassuming and convivial — “a man with four kids never gets to speak at home” — Green meets the JC at an Italian restaurant in the heart of Soho.

He’s obviously more than a regular as we are found a quiet space above the main dining area to conduct the interview and, with delicious irony, successfully request that the house music be turned down for the duration of our conversation.

It is a particularly hectic period for him, given the imminence of BluesFest 2019, a tour with Beverley Knight celebrating the songs of Stevie Wonder and his duties as a domestic chauffeur, ferrying son Albert back to the family home in Amersham from a starring role in the Adrian Mole musical in the West End. He also has some end of year shows with a variety of special guests on Radio 2, another example of following in his late father’s footsteps.

It is hardly surprising to hear Green relate that “music was a big influence from an early age. There was always music on, and I’m not just talking about my dad’s. I had a couple of older brothers and so there was the stuff they were listening to. They were always bringing music into the house.

“So I did get all of dad’s stuff, the Sinatra, the Ella [Fitzgerald] and all the great jazz. But my brothers were into everything from the Ramones to Bruce Springsteen. Without realising it, I got exposed to a wide range of music.”

Did his dad encourage him to pursue a musical path? “No, that’s the strange thing. He never pushed any of it on us.

“But because he worked from home quite a lot, it was always there. If his friends were coming round, obviously they were musicians and there was always stuff going on. Whether he meant to or not, it was a clever way of doing it because we are all obsessed with music — a healthy obsession.

“There was a piano in the house that I always had a bash at. But when I was nine I started the clarinet and then moved to the saxophone when I was 15.”

Having always been into “swing and rock and roll”, he then veered towards acts that used sax players, “people like Bruce Springsteen and the Stones. Unfortunately, there wasn’t too much chart stuff that featured saxophones, so my musical future was in the past.”

Green studied at the prestigious Royal Academy of Music but found that it wasn’t for him. “I just wanted to get out and play. That’s the knowledge you don’t get at university.”

At 19, in another “pinch yourself” experience, he joined the backing band of the legendary Jerry Lee Lewis. He recalls his then self as “pretty clueless”, once travelling to Brazil “without realising how far it was, or how hot it is”.

He was really living the dream. “I remember laughing on stage as Jerry Lee played Great Balls Of Fire. I was so happy.” Another vivid recollection was of playing with Jerry Lee in an unlikely bill also featuring Dionne Warwick and Kylie Minogue at “the first gig in Romania after Communism went down”.

Green toured with Jerry Lee for two years before spending the better part of the decade with Van Morrison’s band. “This will date me [he’s 47] but I wrote him a letter and to his credit, I got a letter back from his tour manager asking me to audition.”

After his father died in 1998, he was back on the road with Jools Holland. “I can’t say enough good things about the man. To have a band that size and keep it going… Jools is a really nice guy.”

One of Green’s fondest memories is of a performance at a Park Lane hotel. “Jools being Jools, he just said: ‘I have a friend who is coming to sit in’.” It was Eric Clapton.”

Via jobs such as artistic director of Ronnie Scott’s jazz club, he also found a niche in radio, initially on Jazz FM.

His Radio 2 career started with Green On Green, in which he recreated the final show his father wrote, and never broadcast, after his mother discovered the script when clearing out some cupboards. It reflected Benny Green’s passion for the Great American Songbook.

For Leo Green, his association with Radio 2 is a labour of love. “It’s an amazing station. In 24 hours, you can hear everything in terms of musical genre.”

However, he describes BluesFest as pretty much a full-time commitment: “We’re already working on next year.”

The inaugural line-up included BB King, whose performance blew Green away. “To stand there and watch him was amazing. I’m still a punter.” Other personal highlights have included Bobby Womack, Van Morrison, Tom Jones, who delivered a barnstorming a blues and gospel set, John Fogerty of Creedence Clearwater Revival fame and Elvis Costello.

The latter came on board having asked: “Is there any chance you can get Georgie Fame as I’m the biggest Georgie Fame fan?” The well-connected Green duly obliged.

For 2019, Grammy-winning John Legend and multi-faceted singer-songwriter Rickie Lee Jones are among the headliners. He credits Legend as having “one of the great voices — he’ll be around for 30 years”.

Of the newer breed, he name checks Jack Savoretti. “I worked with him on a Radio 2 project and was absolutely knocked out. He’s the closest thing to Springsteen in that you feel he’s singing just to you.”

Green doesn’t belong to a synagogue but describes himself “furiously proud about Jewish things culturally, and very vocal about it”.

Israel is a country he has yet to perform in and it’s an omission he is keen to rectify. “There are a couple of good jazz festivals there.”

Other ambitions? “I’d love to have played with Otis Redding and Aretha Franklin but obviously that’s never going to happen.” But he would answer a call from Springsteen or the Stones in a heartbeat.

“It’s a never ending process. I feel like I haven’t got anywhere yet.”

BluesFest is at the O2, London from October 25-27.