'Vi er alle i samme båd,' or, roughly 'we're all in this together'

David Cameron and Sidse Babett Knudsen as fictional PM Birgitte Nyborg Christensen in Borgen (Getty Images/BBC)

A telegenic prime minister holding together a shaky coalition and fighting a battle over independence for a smaller neighbour arranges "date nights" to ensure harmony at home. What could be a description of David Cameron's hectic lifestyle sums up the challenges facing Birgitte Nyborg, the fictional prime minister in the hit Danish TV series Borgen, which has become essential viewing in Downing Street.

The premise of Borgen – Denmark's first female prime minister negotiates coalition politics – is one for connoisseurs of the continental politics of compromise. Yet throw into the mix two beautiful but strident female characters – one of whom finds her secret lover dead in bed – and scandals over extraordinary rendition, election finances and the underhand activities of a wayward spin doctor, and it is easy to see why Borgen has become one of the most talked about TV programmes.

While 800,000 tuned into its debut on BBC4, Borgen is quietly emerging as the iPlayer sleeper hit of 2012, the dissection of high politics and gutter press dominating water cooler and Twitter gossip more than anything Westminster has to offer.

With filming imminent on a new series of The Thick of It – Armando Iannucci's foul-mouthed take on Whitehall's obsession with spin – the British appetite for coalition-inspired drama is at fever pitch, almost two years after David Cameron and Nick Clegg said "I do" in the Rose Garden.

The two coalition leaders insist they have been too busy to tune in – so far – but that hasn't stopped dozens of spin doctors, researchers and civil servants becoming gripped. "I was watching it in the office the other day," said one fan in the corridors of power. "The great thing about the subtitles is you don't need to put your headphones in."

Clegg hasn't watched an episode yet – Danish isn't one of the five languages he speaks – but it's only a matter of time. The Business Secretary, Vince Cable, a fan of Stieg Larsson's Nordic noir trilogy and the previous Danish hit, The Killing, is planning to catch up when he gets the chance. Aides in Downing Street are already gripped. Similarly, Cameron – who was glued to The Killing – will doubtless wait for a long overseas trip to catch up.

Sidse Babett Knudsen, who plays Nyborg, said that she based her character on the way Tony Blair was transformed from a youthful opposition leader to a hardened statesman who took the UK to war in Iraq six years later.

The similarities between Nyborg and Cameron are unavoidable. Last week, we learnt that Cameron not only can remember every minute of his wedding night, but also arranges "date nights" to ensure he spends time with Samantha.

With Nyborg and her unconventionally handsome husband, Philip Christensen, Borgen is slightly more, well, Scandinavian: they arrange to have sex on Tuesdays, until Nyborg is forced to stay overnight in Greenland on tricky diplomatic business. When she telephones to apologise, Christensen says: "Tonight is the night I'm supposed to have sex with the prime minister." Nyborg is trying to placate the leader of Greenland – semi-autonomous from Copenhagen, yet demanding full independence – in a plotline that echoes the tussle between Cameron and Alex Salmond.

Like Cameron, Nyborg frets about putting on weight – in her case, during the election campaign she eventually wins (a victory sealed, incidentally, by the kind of "no notes", off-the-cuff TV that helped to catapult Cameron into the top job). Both Nyborg and Cameron, before winning power, cycle to work and are keen to display their family credentials.

Borgen – which refers to the "castle" of buildings where the PM's office, the Danish Parliament and Supreme Court are based – was broadcast in Denmark in September 2010, four months after the British coalition government was formed. Helle Thorning-Schmidt, who was elected Denmark's first female Prime Minister a year after the first season ran, is married to the son of Neil and Glenys Kinnock and is said to be an avid fan.

While it's little surprise that a political thriller based on a coalition has become a huge hit, it is notable that this one originated in Denmark, not Britain. Each era has had its own defining thriller, but they have all been typically British – until now.

For Thatcher, House of Cards imitated life so closely that it was broadcast over the weeks of her downfall in November 1990. The Major era of back to basics and Tory sleaze was defined by The Politician's Wife, about a woman's reaction to her ministerial husband's sex scandal, while in the Blair years, The Project dramatised a story about Labour apparatchiks and their quest to transform their party and the country.

The most popular drama on British TV last year, Downton Abbey, is set in the past. Yet it seems the best stories are timeless – whether the upstairs, downstairs travails of the Grantham household or the minutiae of Danish politics. It may be just coincidence that both shows began with a secret lover of a central female character dying in his bed. But maybe producers understand that, in any language, sex sells.

The Prime Minister

Coalitions can be a dirty business. When not fighting with smaller partners you must fend off traditionalists in your own ranks worried about immigration and the future of Europe. Thank God for the good-looking family at your side.

Birgitte Nyborg Christensen

Leader, Moderate Party

Little local difficulty: Trouble from the prime minister of Greenland, fighting for full independence from Denmark, over the use of an airbase on the territory.

David Cameron

Leader, Conservative Party

Little local difficulty: Trouble from Scotland's First Minister, Alex Salmond, fighting for full independence from the UK, particularly over the future of defence capabilities in Scotland.

Ousted PM

There's nothing so ex- as an ex-PM. After a tight vote, unpopular incumbents will hang on by chewed fingernails. Borgen's Lars Hesselboe tries biscuits and charm to win people over. Gordon Brown bellowed down the phone at Nick Clegg: "I can't hold on any longer."

Lars Hesselboe

Defeated Liberal prime minister

Tipping point: On eve of the election was revealed to have used a government credit card to buy Mulberry bags and shoes for his wife.

Gordon Brown

Defeated Labour prime minister

Tipping point: Eight days before the election was heard, on microphone, describing a lifelong Labour voter as a "bigoted woman".

The junior partner

Amir Dwian

Leader, Green Party

Consolation prize: Ministry for International Development

Nick Clegg

Leader, Liberal Democrats

Consolation prize: Deputy Prime Minister

Resignations

Every PM hopes for a honeymoon; few get it. For the Lib Dems to lose David Laws after a fortnight over revelations about expenses, was unlucky. Borgen's Michael Laugesen is forced to quit during coalition talks, over an email attacking Islam.

Michael Laugesen

Leader, Labour Party

Time in office: 0 days

David Laws

Liberal Democrat Chief Secretary to the Treasury

Time in office: 12 days

The dutiful spouse

Philip Christensen

Married to Birgitte, two children

Career on hold: Economics lecturer

Samantha Cameron

Married to David, three children

Career on hold: Stationery designer

The scruffy guru

Every PM needs one: for Birgitte Nyborg, it's Jim Royle lookalike Bent Sejro, who teaches her how to "power bluff" through coalition talks; for David Cameron, it's shoeless Steve Hilton, once described by a party aide as a "pint-sized Rasputin".

Bent Sejro

Minister, Moderate Party

Motto: "Power is not a cute little lapdog ... must grab it and hold on!"

Steve Hilton

Conservative Party strategist

Motto: "Let sunshine win the day."

The disgraced spin doctor

Alastair Campbell warned against backroom boys becoming the story. Andy Coulson admitted: "When the spokesman needs a spokesman, it's time to move on." When the boss thinks you're damaging the brand, pack.

Kasper Juul

Press spokesman, Moderate Party

Downfall: Was told to clear his desk after his leader found out he'd helped a rival party to expose the Hesselboes' credit-card scandal.

Andy Coulson

Ex-communications director for Tories

Downfall: Details of phone hacking during his time as News of the World editor mounted. He walked away from No 10, and, soon after, into a police station.

The TV reporter

All that waiting outside government buildings and late-night chat in the green room leads to accusations that political hacks get too close to the people they are supposed to be reporting on. Some closer than others.

Katrine Fonsmark

Reporter and election-debate host, TV1

Dirty little secret: Was having a steamy affair with a government adviser – until he died in bed in their secret love nest.

Nick Robinson

Political editor, BBC News

Dirty little secret: Was president of the Oxford University Conservative Association and national chairman of the Young Conservatives.