The first single from South London indie-synth popsters Bastille was released last year and didn't chart anywhere. Overjoyed 's impact was underwhelming and led to a crisis in confidence for the newly formed band. Their second single, Bad Blood , reached the heady heights of number 90 in the singles chart before scarpering away quickly the following week. Third single Flaws – released last October – went to number 21 and the follow-up, Pompeii, made it to number two.

It°'s the sort of trajectory that used to be relatively commonplace – a band building slowly over a series of releases – but is now virtually unheard of in the quick-turnover pop world, where if you don't hit hard with your very first release, you're commercial history. The slow burn on Bastille paid off in spades last month when their debut album Bad Blood went straight to number one in the album charts.

Live-wise, it's a mirror reflection: last October the band played to a small audience at Dublin's Academy 2. To mark the release of Bad Blood , they were booked for a show in Whelan's for next week (April 4th) but that sold out so quick the venue had to be upscaled to the main Academy venue. That show has sold out in record time.

In The Cinema Museum, a museum dedicated to the history of cinema, hidden away on a south London backstreet, Bastille frontman Dan Smith – a film obsessive, hence the choice of venue for a promo gig – comes across as a reluctant and surprisingly introspective rock-star-in-waiting presence. The consensual bottom line on Bastille is that they are a male Florence and The Machine, a Killers for the pop generation, a band ready to take the next big step up towards arena-ville.

“It’s just one huge surprise after another these days with the singles doing better and better each time and a lot of attention on the album,” says Smith, a 25-year-old with a glittering academic record behind him. “But to be truthful, this world isn’t really me. I was always a quiet and studious type and there are aspects of the music world – performing and doing promo – which I find really uncomfortable to do. In fact, it is my idea of hell, but if it gets the music out there then it's a good thing.”

There are a lot of industry heavyweights in the invite-only audience tonight – eyeing Bastille up for future festival dates – and the four piece don't disappoint with a theatrical set. Smith is transformed on stage – described by the Guardian as "a wild mutation of Coldplay's Chris Martin and Justin Bieber", he looks to the stadium born.

But such was his initial diffidence that he had to transform Bastille from a one-man band to their current group status. “Bastille used to be just me and I was very much in the singer-songwriter mould. The songs used to be very intricate and eccentric – I had written them all in my bedroom – but early on, I decided the last thing I wanted to be was a singer-songwriter. I wanted to be in a band, so Bastille went from just me to the four of us, and I feel much better how we are now,” he says.

Smith has been writing songs since he was 15 but was always too embarrassed to tell his friends. “I just used to write for myself and would never have played them to anyone,” he says. While studying English literature at Leeds University, a friend overheard him tinkering around on one his songs on his laptop and convinced Smith to enter a local talent competition – which he duly won.

“Performing for the first time was a really traumatic experience,” he says. It didn’t help that he describes his early songwriting efforts as “significantly bizarre – quite off-the-wall”. There was also a reluctance, as a solo performer, to write autobiographical songs. What he really wanted was to compose music that reflected his love of cinema. He chose the name Bastille – his birthday falls on Bastille Day – because of his dread of going out under the name Dan Smith.

In keeping with the changing times in the music world, Bastille first got a foothold through YouTube views and the placement of their songs on various TV shows – Hollyoaks, Made In Chelsea – and on a FIFA videogame soundtrack.

While Smith says he’s equally happy writing in the general realms of “indie, pop, folk and hip-hop”, what singles Bastille out from the pack is how – unlike many of their leather-jacked clad indie contemporaries – they aren’t scared of a big pop chorus.

“Pop is important; getting that hook in is important,” he says. “It may not make you the coolest new band around, but I’ve always really wanted each song to be its own story with its own atmosphere – so you need to get something memorable in there.” A synth-heavy sound with minimal guitars ensures Bastille are ripe for radio play, but what surprises Smith most about the reaction to the songs this early in their career is how many audience members knew all the lyrics months before the album came out.

“For someone like me who really isn’t comfortable on stage yet, when you’re playing in a big room and hearing hundreds of people singing the words back to you is a really enjoyable experience,” he says.

With the album going in straight at number one last month and the current single Pompeii just missing out on the number-one slot to One Direction, Bastille are already one of the big breakthrough stories of the year. Even though it's been a slow build for the band over the past three years, Smith says he's a bit shell-shocked by their recent success.

"We made the album in a studio the size of a cupboard," he says. "There was certainly never any expectation that we would even get near the Top 10 with our first album, so all of this feels like it's happening to someone else and we are just watching on. We were on tour with Two Door Cinema Club when we heard Pompeii had gone it at number two, and not really knowing what to do, we ended up in Pizza Hut that night – not the most rock'n'roll celebration ever held. We've been touring so much over the last few weeks that it's hard to get a real sense of perspective, but the fact that so many people bought the album in the first few days of its release was a great joy."

With so much indie landfill out there at the moment – identikit guitar rock going nowhere fast – what has lifted Bastille up is their mix’n’match sound and their popularity with the crucial early twentysomething record buyers. Smith speaks of his great appreciation of soul and hip-hop and of how he grew up listening to Jurassic 5 and Crosby, Stills and Nash. But it’s his A-Ha-like sensibilities which give his songs a pop sheen.

As happy talking about the musical merits of Damon Albarn, Kanye West, Bon Iver and Jack White as he is confessing to a chronic obsession with film-maker David Lynch (there's a song on the album called Laura Palmer ), this is also a man who is happy finishing a Bastille show with a sincerely done cover version of Snap's euro-dance tooth-rot anthem Rhythm Is A Dancer .

“You can say what you like about that song and I’m still really not sure about it, but the one thing it does have is massive hook. A bloody massive hook. And that’s good enough for me,” he says.