"Where's my plane? Eh?" says Wolf Alice bassist Theo Ellis, in mock indignation.

The Londoner, who in 2012, along with drummer Joel Amey, joined the existing duo of Ellie Rowsell and Joff Oddie, who'd been performing together for two years already, didn't imagine being in a rock'n'roll group was going to be like this.

"I used to watch bands in my teen years and think, 'Look at them on stage, I've bet they've got about 12 planes each'. It's not really like that," he admits, defeatedly.

Singer Rowsell, who has now performed on just about every significant stage in the country, has been equally shocked since Wolf Alice started gathering momentum three years ago.

"I would watch bands at places like Shepherd's Bush Empire," says the 23-year-old, "and they'd look so together and composed, and not for a minute did I think there might have been nerves before the show.

"I just thought they'd be like, 'Another day, another gig', and completely self-assured that everything would run smoothly.

"I am so scared going on stage, thinking about the people I know in the crowd and all sorts of things, and all those other bands must've been too."

If she is scared, she hides it well, commanding a presence in the centre of the stage with her big voice, sweet one second, a throaty howl the next.

After the line-up was completed three years ago, the band released their first singles and EP, which immediately put them on the map with early tastemakers, such as Radio 1's Huw Stephens and NME magazine.

The wave of bands they came through with - Swim Deep and Peace, to name two - have all released debuts and either have second albums out already, or ready to be released soon.

There is, however, something pleasingly old-fashioned about the way Wolf Alice took their time, toured the country and got better and better as a live band before attempting their debut.

If they had any plan, it was to have no plan, and to say yes to any gig offered, whether it was put on by "the biggest idiot, a really cool, super-chic fashion house or something really naff".

"We just wanted to play," says Rowsell. "We said yes to everything, but aside from that and the whole, 'We've recorded a song, let's release it!', we never thought long-term about where it was all going."

She, modestly, doesn't think the band have changed that much since they first started, but on reflection, concedes they probably have.

Drummer Amey adds: "We don't realise how much we've changed, I don't think. We've come so much further than we realise."

That's quite an understatement, considering only two months ago they were in a chart tussle with Florence + The Machine. Flo, who was headlining Glastonbury that week and received a huge sales bump as a result, won out, and her third album stayed at No 1, but Wolf Alice's debut getting to No 2 is, if anything, a bigger achievement.

"And it leaves us room for next time, right?" says Rowsell, trying to hide her disappointment.

"People were coming up to me at Glastonbury telling me we were at No 1," says Ellis, "and we were No 1 in the midweek charts, but Florence is hardly a bad person to lose out to."

Regardless of where it charted, it's a special record.

There's a high-sheen gloss to My Love Is Cool, which, rather than jar as it does with many young guitar bands who attempt such high production values, it suits Wolf Alice to a tee, giving the album an immediately familiar, classic feel, reminiscent of, in different places, The Breeders, Pixies, Lana Del Rey and My Bloody Valentine, but always unmistakeably Wolf Alice. It's quite an achievement to carve out such an identity in the face of such lofty influences and shifting musical styles.

"I suppose that's just because it's us playing, recording in the same room with the same producer," says Oddie, almost dismissively. "Maybe if we'd have moved around with different studios and producers, it might not have that shared sound?"

Not buckling under the weight of expectation is another thing they should pat themselves on the back for. Few bands resist demand for a debut record for the best part of two years, and then deliver an album as assured as this.

"It was scary, actually," says Rowsell. "We built up a lot of momentum and people were talking about us before the record was made, but we worked hard not to let that change anything. As soon as we made it, we felt really proud of it. And we really like it ourselves, which is obviously important."

For now, as they gear up for six months of touring, kicking off with Australasia, Reading and Leeds, followed by a string of UK shows, Wolf Alice have jumped their first hurdle with ease, and have given themselves a target for their second album.

As for fame, "there's no downside", according to Ellis. "We feel like a proper band now we have the album," he says. "We have something to show the grandkids, anyway."

If they carry on like this, it won't be long til they get those planes.

EXTRA TIME - WOLF ALICE

:: Wolf Alice take their name from a short story by Angela Carter, about a young feral girl who cannot be civilized, despite the efforts of the nuns trying to tame her.

:: They released their first full EP, Blush, in 2013.

:: The band recorded their debut album in Wood Green, North London. Ellis grew up nearby, and his memories of the area involve being chased around as a teenager. The rest of the band, meanwhile, have more fond memories, such as the kebab shop they frequented during recording.

:: They're currently on a tour of Australia, New Zealand and Japan, prior to their UK dates next month.

:: Their song, Moaning Lisa Smile, was used during an episode of HBO series The Leftovers.

:: Wolf Alice's debut album My Love Is Cool is out now. They play Reading and Leeds Festival between August 28 and 30, and begin a UK tour on September 15. Visit wolfalice.co.uk