Jason Williamson decides the best place for our interview is a central Nottingham branch of Caffè Nero, where we start talking about the kind of music perhaps a little too soaked in classicism and pastiche for its own good. There follows a brief exchange about Miles Kane, the Wirral-born, Arctic Monkeys-associated, mod-pop solo artist– much loved, as it turns out, by the woman who serves us. With a look of amused mischief, Williamson then reaches for his phone, and shows me a tweet he fired off to Kane when the latter bigged up Williamson's group, and tried to become one of their followers. It informs Kane: "This music was born out of a hate for pretenders like you. You can either leave gracefully or I will block you."

As an example of Sleaford Mods' essential view of the world, this is illuminating – though so are no end of other comments on the band's Twitter feed (from this year's Glastonbury weekend: "I thought the dog had shit in my hair but then realised Kasabian were fucking playing"), and the range of targets in Williamson's lyrics: among them Kate Moss's husband – Jamie Hince of the Kills, Oasis, a former member of the acid house-era flash-in-the-pans Candy Flip, and no end of bêtes noires known only to him and his friends. I ask him an obvious question: are his words and indication of what it is like in his head? "The rage?" he wonders. "Yeah. That's why Sleaford Mods is. Definitely."

In person, Williamson – who is soon joined by his creative partner and the creator of Sleaford Mods' music, Andrew Fearn – is an engaging, loquacious and repeatedly funny presence. Besides, to focus too much on the gobbets of hate and disdain scattered through his work is to risk doing him and his group a disservice. More important is how powerfully the band's work speaks about the condition of the country and life at the economic arse-end, and the amazing originality of what the two of them are doing. Williamson is 43 and Fearn is 42, and both have a long line of music projects behind them – which, in the sense that Williamson in particular has had a bellyful of what some people call retro-rock, may explain the sense of an inspired leap into virgin creative territory.

Their admirers in the media have cited reference points such as the Fall, John Cooper Clarke, and the punk-era New York avant-gardists Suicide – all comparisons Williamson and Fearn insist are coincidental. The fact is that in an age where borderline plagiarism sometimes feels pretty much inevitable, Sleaford Mods sound like nothing else, from either the present or the past.

To describe their sound is not easy. Fearn's music is unbelievably sparse, sometimes stripped down to just bass and drums, usually clattering along as if to evoke a mind in overdrive. Williamson's words – delivered relentlessly, and in his chewy East Midlands accent – often flip between direct descriptions of life's most rage-inducing aspects ("I worked my dreams off for two bits of ravioli and a warm bottle of Smirnoff/ Under a manager that doesn't have a fucking clue"), stream-of-consciousness tirades that conjure up a world of sticky cafe table-tops, grinding hangovers, and the cold comfort of consumerism ("Weetabix, England, fucking Shredded Wheat, Kellogg's cunts") and the odd political statement, with either a large or small "p": "I can't believe the rich still exist, let alone run the country"; a reference to "the lonely life that is Tory".

Williamson is from Grantham, the birthplace of Margaret Thatcher; Fearn was born near the Lincolnshire town of Saxilby. Both sets of parents did well out of the up-by-your-bootstraps ethos of the 1980s and founded successful small businesses but – in different ways – their sons eventually decided to take an alternative path. Fearn was a "moody Smiths fan" who shocked his dad by becoming a vegetarian when he was 15; Williamson decided the Thatcherite dream hadn't worked out when he was in his mid-30s, and "started getting shunned by people because I couldn't hold it together – I was doing agency jobs, and being frowned upon, and that's when I started to look at things differently."

Williamson's progress through his 20s and 30s was not easy. He spent nine abortive months trying to find musical soulmates in San Francisco, came home and played guitar and sang in a succession of bands, and ended up trying to ride the wave of late-period Britpop in a group who were almost called Sunday Dinner, before settling on Meat Pie. Their aim was to somehow combine Small Faces and Guns N' Roses, but cocaine did for them. "Meat Pie finished 'cos I got into sniff and started acting like a right arrogant bastard: people telling me I had this great voice, and I should be signed," Williamson says.

At certain points, he says, his coke habit "was getting stupid." How much was he doing?

"Not a lot. Probably about three or four in a night. Grams."

That's a reasonably prodigious habit.

"I still don't think it's a lot. My wife was like: 'You're a fucking twat.'"

"All my money was going on that, and I wasn't eating. Just drinking, and doing coke, and working. And getting sacked from jobs, and going from job to job. And that's when the psyche of Sleaford Mods started to form. I was taking note of the fact that I was failing, and taking mental snapshots of it: remembering all the times that I fucked it up and went even lower. That started to build up, and build up. I started to get curious about why I was doing it. I started to view the situations I was in as inspiring."

The eureka moment came one morning in spring 2006. "I had no money. I'd just have enough for a Mars bar, most days, and a can of Special Brew. And I wrote a song called Teacher Faces Porn Charges, about going to the shop in my pyjamas, to buy the Mars bar and the can."

A friend, Simon Parfrement, (nicknamed "Parf"), and still an integral part of Sleaford Mods' set-up) suggested combining Williamson's words with a loop lifted from a Roni Size record. "And it worked, straight away," he says. "It was better than anything I'd ever done. I took it home, and I couldn't believe how good it was. That's how Sleaford Mods was born."

The new project was originally called That's Shit, Try Harder. The name Sleaford Mods came from a Lincolnshire town 15 miles away from Grantham and the subculture into which Williamson had jumped when he was a pre-teen fan of the Jam, though Paul Weller is among those now disowned, Williamson told an interviewer: "He's just going through the motions. He probably just sits back and thinks about his back catalogue but that's no excuse. I feel really strongly about stuff like that because he shouldn't be doing it anymore.". By way of more modern touchstones, a work colleague had turned him on to the Wu-Tang Clan, and his ears pricked up at the Streets' debut album, Original Pirate Material. As Williamson found his voice and felt newly liberated from rock music, there followed four Sleaford Mods albums, pressed up piecemeal, before Williamson resolved to up the ante and recruited Fearn after hearing him play at a Nottingham bar called the Chameleon.

Unwittingly, his new partner had prepared himself for the Sleaford Mods' world by working for four and a half years cold-calling people about gym membership ("Mainly calling old people who could barely walk"), and had already seen Williamson in action. "I thought it was awesome," says Fearn. "Friends of mine hated it. They didn't know how to take this shouty man. I just thought it was fucking brilliant."

Which brings us, via Wank (2012), Austerity Dogs (2012) and this year's acclaimed Divide and Exit, to now. It seems almost comically apt that Williamson works as a benefits adviser for Nottingham city council but he will finally turn professional in October. Lyrics for new songs are already piling up in an app on his iPhone. With the help of the man who puts out their records – Steve Underwood, who works in Nottingham city centre as a bus driver, and joins us in the cafe and then a nearby pub, along with "Parf" – next year, he and Fearn will tour an array of off-the-tour-circuit towns where they think their music will make perfect sense: among them, Wakefield, Scunthorpe, Hitchin and Ramsgate.

Bizarrely, they already do well in Germany; where, says Fearn, "people go fucking mad". Their regular hotel in Hamburg inspired the priceless opening line of the song Tied Up In Nottz: "The smell of piss is so strong it smells like decent bacon" and Williamson recently made sure a picture of the toilet in question was tweeted. "It still smells the same," he says. "Terrible."

Toilets and human waste are a common theme in their songs. Why all the scatology? "It's just such a horrible thing," says Williamson. "I still can't get used to faeces. It's just fucking horrible. I mean, mine's disgusting. It's inspiring: having to go to the toilet and release all of it. And then you walk out of the loo and there's a picture of Beyoncé looking stunning. You can't connect the two things, can you?"

•Sleaford Mods play the Lexington, London, on 18 July. Divide and Exit is out now on Harbinger Sound