'Do you know Dennis and Lois?" asks Justin Young, singer, songwriter and guitarist of the Vaccines. "They used to do merch for the Ramones. They're these New York legends." Dennis and Lois, it turns out, have been telling the Vaccines their theories about Johnny Ramone.

"Johnny hated wearing his leather jacket. He saw himself as a businessman and the leather jacket as an opportunity to create a brand. Offstage, apparently, he had been wearing a check shirt and big glasses. He kept all his money in cash and hid it – Dennis and Lois used to housesit for him and he just shoved all his cash in his records. They don't think he's dead. They're convinced they saw him having a curry. In his favourite Indian restaurant in New York. Wearing exactly the same clothes."

"If Johnny Ramone were to fake his own death," bass player Arni Hjörvar observes, entirely reasonably, "I'm sure he'd be clever enough to leave New York."

It's a bit of a relief to meet this loquacious version of Justin Young. Journalists meeting him last year – when the Vaccines were exploding in a way received wisdom suggested guitar bands no longer did – often reported him to be shy, defensive and unhelpful; one review of their first album was split between excoriating Young as a person and celebrating his songcraft. Even the Vaccines' publicist, without having been asked, offers reassurance that he is a much happier interviewee these days.

Today, at the BBC's Maida Vale Studios, where the Vaccines will later record a live session for Zane Lowe's Radio 1 show in front of an audience of competition winners, Young is both confident and charming, and he's obviously been boosted by the group's success: "It scares me how easy I find songwriting … I know we're a great band … I have more focus than I've ever had in my life." Even so, he manages to come off as puppyishly enthusiastic rather than an arrogant bore. Evidently he finds being a rock star easier than he did becoming a rock star.

The Vaccines made their London debut in October 2010 at a tiny free show. By the following April, they were headlining two shows at the Electric Ballroom, 150 yards down the road from that first gig, to 1,000 people a night, among them the boys in rugby shirts who throw their pints in the air and the teenagers who decide you are Their Band, and whose patronage turns a band from a cult to a sensation. Come the end of 2011, it was two nights at the 5,000-capacity Brixton Academy. By the start of 2012, their debut album, What Did You Expect From the Vaccines?, had gone platinum – just as the world was being told guitar music was dead.

"I had always looked down my nose at bands I'd seen accelerate too quickly," Young says of the band's ascent. "I always said: 'Oh, I'm never gonna build my house on sand.' But it was absolutely impossible to stop."

"At the time we were trying to stay in control of it," Hjörvar adds, "and I think we felt we were in control. But it feels very different now to how it did then."

"For someone who's insecure, it's quite an uncomfortable prospect, the idea that people who have never met you and never will are talking about you, not just as a musician, but as a human being," Young says. "I think we were all – me in particular – quite uncomfortable, and we were just doing everything we could to slam on the brakes."

But if they had been able to slam on the brakes, though, they might not have made it here. The fate of any band that generates as much buzz as they did is to be dropped unceremoniously by the anointers of cool. The challenge is to win a wider audience in that time, which is where the beer throwers and the teenagers come in. "There was this realisation," Hjörvar says of their new audience, "and I said: 'Justin, we're not a cool band.'"

"There was a sudden change, wasn't there," Young says of the band's fanbase. "It was quite an uncomfortable realisation when we realised our fans probably aren't really like us. But you know what? It feels a lot better playing to a room full of people who are happy to throw their beers up in the air than it does to people standing with their arms folded, tapping their feet."

Out of that ascent has come one of the lead singles from the second Vaccines album, The Vaccines Come of Age. Teenage Icon appears to be Young's mea culpa to those who accused him of being, well, a bit boring, with him singing of being "reserved and shy/ Your average guy/ No piercing stare/ Just out of shape with messy hair". In case you hadn't guessed, he is actually "no teenage icon". There won't be any more songs like that, Young promises. "I really hate tour albums or band albums where people are singing about that." He croons a snatch of Kiss's babe-it's-lonely-on-the-road ballad Beth. "How can anyone connect with that emotionally? So Teenage Icon was almost a mistake."

It's difficult to argue that he is trying hard to resist teenage iconhood now though. The Vaccines have even adopted a uniform – from 50 yards away, when I first see them on the street outside the studios, they could not be mistaken for anything bar a rock band. All except Hjörvar are wearing denim top and bottom, with Young and lead guitarist Freddie Cowan sporting jackets with the sleeves ripped off. Drummer Pete Robertson, who has kept his sleeves, is so hirsutely transformed I mistake him, embarrassingly, for someone else entirely.

"The way bands look has always for me been a really important part of pop and rock'n'roll music," Young says. "If a band looks good, they're more seductive. It wasn't supposed to look heavy rock. Double denim is symmetrical and timeless and …"

"Easy," offers Hjörvar.

"Yeah."

"It means you can travel with a tiny suitcase."

Young tries to direct the train of thought back to rock'n'roll. "It does make me feel like I belong to something when all of us are wearing the same thing."

The moment Young knew the band had hit on something was with the song If You Wanna, which circulated online as a demo in summer 2010. "When we started, it was just a bit of fun," Young says. "We were all a bit depressed. But a mate had a studio and we had a deal that we could go in and use it for free. I thought we were gonna stare at our feet and make 10 minutes of noise. That was the plan. We did If You Wanna and I sang the chorus as a joke, because it was cheesy, but everyone was: 'That was fucking cool.' That's when we thought: why not be a pop band? I remember thinking: 'Enough people are gonna like this that we're gonna be able to headline the Barfly. This is a fucking great pop song.'"

They did like it. And then, as is often the way of things, came the backlash. Young's lyrics were too clunky, some said ("I like stupid fucking clunky lyrics," he responds. "People probably don't realise how knowingly dumb they are"). The music was basic and reductive ("All the pop music I like is young, dumb and primitive. I want to make that kind of music"). They were artificially inflated by being on the BBC Sound of 2011 shortlist ("Look back on the other bands who were on the BBC shortlist the year we were. It wasn't exactly the key to the city"). And they were just braying poshos, part of an aristocratic takeover of British music ("Let's be honest, we don't have anything in common with those bands we were being compared to").

Young tried to avoid reading too much of what people were saying about him. Not because he doesn't care; he says he finds it too hurtful. He and Hjörvar both bemoan a culture that has turned everyone into critics, and which puts bands under intense pressure to deliver something spectacular long before they are ready. "There's a hunger to have something new every day," Young says. "Guardian New Band of the Day is the perfect example of that. And so bands are in and out of vogue quicker than they have ever been before. If you're being talked about before release and you don't have a record's worth of songs then you've fucking blown it before you've begun. And blogs in particular are to blame for that. Every day they have to post a new band. Are they still listening to the band they posted three weeks ago."

The Vaccines, however, did have a record's worth of songs, having hidden themselves from view until they did so. "I'd seen people just fall off cliffs," Young says. "And our thing was: I don't want anyone to hear this until we're ready for people to hear it."

A couple of years on and the Vaccines have not reinvented themselves so much as turned on the spot, to face 90 degrees around from their previous orientation. It's still punky and – Young's favourite musical self-definition – primitive, but the musical accents are slightly different, harking more to their cover of the garage standard Sometimes Good Guys Don't Wear White than the Ramonic tendencies of their debut (their version, in a deliberate nod, was actually a cover of a cover – of Minor Threat's version rather than the Standells' original). More importantly, the reverb of What Did You Expect has been stripped away, so you can actually hear the band beneath the echo. Cowan, especially, sounds great. That's even clearer when they play live in the studio – with Young singing straight to the desk and not coming through the speakers, you can hear Cowan's supple, imaginative guitar lines clearly. "Shredding!" exclaims Zane Lowe, never knowingly undersold on hyperbole.

You can also hear just how simple and timeless the Vaccines sound is: guitar, bass and drums locked into a pattern that sounds immediately familiar but also – as with the best bands built on the three-chord tradition – timeless and new. "It keeps me awake at night because so much comes," Young says of his songwriting. "When we were putting the songs together for the record I had sleepless night after sleepless night because we had too much stuff and I was like: 'What if there's a Baby, I Love You or a Maggie May I've just discarded."

"Justin brought in a good 160 songs all together," Hjörvar says.

"We could do 169 Love Songs. Quite often I'll see an interview with a band and they'll say: 'We had 50 songs,' and I think: 'Is that the best you can do?' I really hope people don't think that about us."

Now he is convinced of the Vaccines' greatness, Young has a new worry to contend with. What happens when you stop being great? "I really like the idea of giving it up before it all goes downhill," he says, gravely. "But I also quite like it when bands ride the storm – they fade into obscurity and then come back. I can't think of a band who've been dignified over a whole career, to be honest. I like seeing people grow old with dignity. I guess someone like Nick Lowe has become more dignified as he gets older. As soon as you are too old for pantomime, you shouldn't really be taking part in pantomime. But it's much easier said than done, isn't it?"