The broken clock had perfect timing.

“Twenty Twelve,” a satirical BBC sitcom about the London Olympics, kicked off with an episode that saw organisers unveil a baffling backwards-running clock counting down to the start of the games.

The same day, a real clock was installed in London’s Trafalgar Square to tick down the 500 days until the July 27, 2012 opening ceremony. The next day, it stopped dead. It was several hours before engineers got it going again.

For Olympics planners, it was an embarrassment. But it was priceless free publicity for the show, which has become must-watch television for those involved with organising the games — and anyone who has ever wondered whether London’s creaky infrastructure can cope with an Olympic-sized challenge.

“It was delicious,” the show’s writer and director, John Morton, said of the timing. “Like so many things in broadcasting, it was a complete accident.”

“Twenty Twelve” has had luck going for it, as well as considerable charm.

In the mock-documentary format pioneered by comedies like “The Office,” the show goes behind the scenes at the fictional Olympic Deliverance Commission, run by long-suffering “Head of Deliverance” Ian Fletcher (Hugh Bonneville, recently seen as aristocratic Lord Grantham in costume drama “Downton Abbey”).

Each week is an exercise in Murphy’s Law: An athlete hired to help whip up excitement about the games turns out to be phenomenally boring. London’s traffic chaos plays havoc with plans to give Brazilian delegates a tour of the Olympic site. Roman remains are found under the diving pool — an echo of a real-life incident, in which the remains of an Iron Age settlement were discovered on the aquatic centre site in 2007.

Each indignity is endured by a team of well-meaning but befuddled managers and a perky “Head of Brand” (Jessica Hynes), who spouts a torrent of PR bafflegab: “Here’s where we ramp up the public interest and take it to the next level ... go viral and launch 2012 2.0.”

Looming offscreen are the real-life presences of London Mayor Boris Johnson — depicted as a wildly talkative bumbler usually found on a bicycle — and London organising committee chairman Sebastian Coe, the best-known public face of the games.

Coe even makes a cameo appearance in the second episode — though he’s modest about his acting impact.

“I don’t expect to be seeing a BAFTA (British Academy Award) heading my way anytime soon,” Coe said.

Like Coe, other Olympic insiders seem good natured about being sent up on prime time TV.

The London organising committee, known by the acronym LOCOG, said the show was “becoming compulsory viewing for the staff every week.”

“We hope we don’t have too many life-imitating-art experiences,” said a spokesman, who was not authorised to give his name for publication.

He added that the show is “a great example of the British sense of humour that we hope visitors to the games will experience next year.”

Morton, whose previous work includes the mock-interview show ”People Like Us,” thinks there is something essentially British about the dogged character of Fletcher, who “carries on with this relentless optimism in the face of all evidence to the contrary.”

The show’s makers were not given insider access to Olympic planning.

“We didn’t have any spies in the camp,” Morton said. Instead, he drew on the knowledge that all large organisations are dysfunctional in similar ways, and the gap between Olympic organisers’ noble rhetoric and the messy everyday reality of planning such a big event.

Morton said the comedy lies in “the yawning gap between the high-sounding public pledges and the day-to-day nitty-gritty details — how should we sequence the traffic lights? How many toilets should there be in the Olympic village? There is comedy to be had in that disjunction.”

The result, he said, is not meant to skewer real individuals.

“It’s not out to destroy any one person or institution,” Morton said. “It’s a parallel universe and I hope it’s a benign satire rather than destructive.”

The six-part “Twenty Twelve” has two more episodes to run. It has been getting decent audiences on the broadcaster’s highbrow channel BBC4 - and, crucially, good word of mouth, although not everyone is a fan.

Creators of “The Games,” an Australian comedy broadcast in the run-up to the 2000 Sydney Olympics, have accused “Twenty Twelve” of pilfering their idea, saying they pitched a London-set show to the BBC in 2006 and met with Morton to talk about it.

The BBC acknowledged it had had contact with “Games” producer Rick McKenna, but said “Twenty Twelve” was “an entirely original series.”

“No use has been made of any material deriving from The Games and we are confident that the allegations are without foundation,” the BBC said in a statement.

Morton hopes there will be a second series, so he can raise the temperature under his characters as the 2012 opening ceremony draws closer.

“It will be the same people but under even more pressure because it’ll be too late to change anything,” he said.