Alabama 3 are discussing dead gangsters, both real and fictional. This year the Brixton band lost James Gandolfini, the American actor who played the mafia boss Tony Soprano, and Bruce Reynolds, the British brains behind the Great Train Robbery. The Gandolfini connection comes via Alabama 3 providing the theme tune to The Sopranos, while the Reynolds connection is that Bruce’s only son, Nick, is a member of the band. “It is a strange coincidence,” Nick Reynolds admits, “and not one I’m given to dwelling on. I think my father chose crime because he grew up on Second World War stories of action and heroism, and was looking for excitement. There was nothing thuggish about him. He was an intelligent man, well read, debonair. But he made some insane decisions that would have huge repercussions across our lives.”

Nick Reynolds’s life has seen him work as a diver for the Royal Navy during the Falklands war, then develop into an acclaimed sculptor — his work will be included in a V&A exhibition devoted to protest art next July — and succeed as a musician. As a member of Alabama 3, he plays in the best British rock’n’roll band since the Pogues, or quite possibly the Clash. Yet most people know them only for the Sopranos theme.

“It’s a blessing and a curse,” says Rob Spragg, Alabama 3’s lead vocalist and main songwriter. “Being on the show has obviously benefited us, and meant that wherever we go in the world, we have an audience. It’s like having a hit single that keeps on giving. But we are much more than the song Woke Up This Morning.”

Did the band get to know Gandolfini? “When the series first came out, we went to New York,” Spragg recalls. “He was a good guy. Loved to party.” He chuckles at the memory; Alabama 3’s reputation for extreme hedonism suggests that common ground was easily found.

So what did Bruce Reynolds make of The Sopranos? “I’m not sure he ever watched it,” Nick replies. “Having moved in that world, he may have been immune to how TV and the movies portrayed gangsters. That said, my dad modelled himself on Cary Grant in To Catch a Thief — the gentleman thief. He was something of a fantasist, in the sense that he wanted to steal money without taking from anyone who would miss it or involving any violence.” He then quietly adds: “We were very close. I miss him immensely.”

So close that Bruce even appeared on Alabama 3’s 2005 album Outlaw: they recorded the 1960s folk song Have You Seen Bruce Richard Reynolds? and got the veteran villain to add a few words. “That was fun,” Spragg says. “Bruce was a fascinating character to be around.” Alabama 3 performed Too Sick to Pray, a beautifully mournful blues song, at Reynolds’s funeral in March. “It was a nerve-racking gig,” Spragg says. “See, the way the church was set out, we were facing the front pew, which was packed with some heavy characters. I could see them all staring at me and thinking, ‘Who is this idiot?’”

He laughs, then admits that being associated with The Sopranos means the band inevitably meet real-life gangsters. “They all want to buy us champagne, and are completely unaware I wrote Woke Up This Morning as a result of reading about Sara Thornton, who, after years of suffering domestic violence, killed her abusive husband.”

This ragged collective brilliantly blend music with witty polemics. “Our debut album came out in 1997,” Spragg says, “right at the height of Britpop. We were mixing gospel and blues and country and acid house, and talking about Tony Blair being subservient to America, the opposite of flag-waving bands like Oasis, so we got shafted.” To this day, Alabama 3 remain outsiders, loved by a loyal fan base — including the authors Stephen King and Irvine Welsh — but rarely given the critical kudos they deserve.

“They say Americans don’t get irony,” Spragg continues, “but what do you make of idiots at the NME who don’t understand that Alabama 3 stand as a critique of how Americanised Britain is? I grew up in Wales, with a Mormon minister father, and heard gospel at the temple on Sundays. Every UK high street has a McDonald’s and a Subway. From the Beatles to Dizzee Rascal, British musicians have always been in thrall to American music. It goes on and on.”

“My father’s an example of how seriously the British embrace America,” Reynolds adds. “Hollywood movies and jazz shaped his identity. Ronnie Scott’s is where he’d do his deals — not in some East End pub.”

Still, the band’s penchant for addressing audiences in accents normally found in the Deep South, while meshing steel guitar and harmonica over pumping dance beats, has led to a certain amount of bafflement. “We’re not easy to categorise,” Spragg admits. “We were signed to Sony at one point, and the guy from the label said to me, ‘Rob, tell me how I market you. Are you rock? Or country? Or dance?’ In the Welsh valleys, the miners loved to party to country music from Nashville, just like their kids now party to Chicago house.”

Indeed, his best songs are equally effective whether played with storming beats or by an acoustic Alabama 3. “We’ve always been ambitious musically and conceptually, and, all right, we confused some people. But I’m hoping now that dance music is huge in America, and DJs are starting to add banjos to their mixes, and kids are so eclectic in what they listen to, that there might be more understanding of what we’re doing.”

Alabama 3 are widely regarded as one of the world’s best live bands, and touring is what keeps them afloat. As they prepare for December dates, their headquarters above the Jamm Club, in Brixton, bustle with activity. The band now run their own label and hope to operate as a mini Motown in south London: already they have used their studios to nurture and record talent from the surrounding council estates. “The kids come in here and they know how to rap, but beyond that they’ve no idea about making music.”

Spragg and Reynolds are also involved in several film projects — acting, producing, soundtracking — while Alabama 3 appear in a forthcoming Irish feature, Songs for Amy. “We have a real work ethic,” Spragg says. “No matter how hard we party, we’re always ready to take it to the stage.”

With a bemused grin, he then admits: “We recently received a certificate of commendation from the Mayor of Lambeth for our community activities. Whoever would have thought that a bunch of old reprobates like us would receive official acknowledgement?”