When London-based singer/songwriter Dan Smith called his band Bastille, he was merely thinking of his birthday, July 14, France’s Bastille Day.

But for the biggest-selling new British act of the last year, with hindsight Dan’s choice seems an ominously apt metaphor for their dramatic impact.

With debut album Bad Blood entering the UK charts at number one and picking up a prestigious Brit Award earlier this year, Bastille are building on that success with a raft of top festival slots, including this weekend’s V Festival in Chelmsford.

Dan says: “We kind of went into the Brit Awards completely baffled by the fact we were there at all. We felt a bit out of place, so we mainly concentrated on the fact we were performing at probably the most public thing we’d ever done.

“We weren’t expecting to win at all, so the shock on our faces was pretty genuine. We were in a bit of a daze really, and wandered up and had to cobble together some semblance of a speech, and then went off to go and get ready to play.”

Following the Brit Awards, their album Bad Blood went back to number one again a year after it was first released, and so it now is one of the biggest-selling albums of 2014.

“It was mad,” Dan adds. “Especially as it stayed at number one for two weeks. You know what, it was a really great feeling and not something we expected. With the Brit Awards, it’s such a massively public thing, it gets a lot of attention and as a band we’ve always just got on with our own thing relatively under the radar.

“We’ve always done alright, but we don’t get a huge amount of publicity, so it was interesting doing something like the Brits. I think it meant people who had maybe heard one or two of our songs over the course of last year could finally put some faces to the name and maybe that’s why the album thing happened.

“Being involved in something like that was pretty surreal and a lot of fun for us. Getting to see people like Arctic Monkeys and Beyonce play, doing our tune with Rudimental, all of it was pretty weird and awesome.”

Bastille was formed by Dan in 2010 after recruiting keyboard player Kyle Simmons, bassist Will Farquarson and drummer Chris “Woody” Wood.

While they only pressed 300 copies of their 2011 independent debut single Flaws, its accompanying video, edited by Dan using clips from Terrence Malick’s 1973 cult classic Badlands, scored half a million hits on YouTube.

Signed by Virgin Records and tipped by a vociferous network of discerning bloggers, after three singles they were selling out their first headline UK tour before their album was even released.

“It’s weird because we never discussed any big ambitions,” he says. “With that tour, when we sold out two nights at Shepherd’s Bush Empire, we thought, ‘Wow! This is brilliant!’ I don’t think we ever imagined it getting any bigger than that.”

But it did. Infectious fourth single, the anthemic Pompeii, charted at number two. It went on to become the second most-streamed track of 2013, just behind Daft Punk’s Get Lucky, and holds the record as the song to spend the most number of weeks at number one on the Official Streaming Chart.

The album, Bad Blood, followed, smashing in at number one and quickly achieving platinum status in the UK. The most downloaded album of 2013, and the second most-streamed, it’s since sold more than two million copies worldwide.

So Bastille shouldn’t really have been surprised at last year’s Glastonbury Festival when they drew the largest-ever recorded crowd in the history of its John Peel Stage.

“You can be told all these sales statistics, but they’re all abstract,” Dan adds.

“It’s only when you play live that you feel it, seeing that reaction among our audience. Those are the proper markers of success. Glastonbury was truly incredible.”

Ask Dan to pick a highlight from Bastille’s whirlwind rise and he’s spoilt for choice.

Possibly the honour of being the first band ever to play the British Museum when invited to perform Pompeii at the opening of its Life & Death Pompeii & Herculaneum exhibition.

“It was just strange serendipity that the week it opened there happened to be a band in the charts with a song about Pompeii,” he jokes, “so they invited us along.

“We had to sing before the leading archaeological minds, right beside the ancient relics.

“We were thinking we shouldn’t be allowed to do this, but the moment we started singing it just felt really nice. They ended up asking us to sing it twice.”

Or his meeting with his all-time hero, David Lynch, whose 1990 TV series Twin Peaks inspired Bastille’s Laura Palmer and who asked the band to remix the track Are You Sure from his 2013 album The Big Dream.

“I was very nervous,” Dan says, “but he was so nice. He just stuck out his hand said ‘Hi, I’m Dave. You must be Dan?’ and my head exploded. David Lynch is the biggest rock star in my world.”

Next up for Dan will be playing this weekend’s V Festival.

He says: “We played V Festival a couple of years ago as a completely unknown band in the Strongbow tent. It’ll be fun to go back and hopefully play to a few more people this time round. We’re not doing that many festivals or shows in the UK this year, so we’re looking forward to V.

“After festivals, we’re taking a few weeks out for Woody to get married and then to properly record the main chunk of the new album.

“We’re also working on another mix tape at the moment, so we’ll be getting that ready and doing a whole bunch of writing and recording. We’ll also be heading back to the US for quite a big tour later in the year, which I’m excited about. I can’t really believe the size of the venues we’ll be playing – it’s blowing my mind a bit.”

Bastille play the V Festival at Hylands Park, Chelmsford, on Saturday. Visit vfestival.com

Make sure you get tomorrow's (Friday) paper for our special preview of the weekend event.