Day four of Extinction Rebellion, which has been a phenomenal success, as climate protesters occupy four pressure points in Britain’s congested capital. Everything has conspired to maximise attention for their climate cause, with the news unnaturally quiet in this pause in Brexit hostilities.

Compared to Brexit rage and fury, climate protesters are the soul of peace and goodwill, their pink boat welcoming Oxford Street shoppers, who mainly stop to talk. The surprisingly few police are friendly, with the 400 arrests curiously voluntary, people well-warned in advance so those who can’t be arrested step back. Is it true the police cuts have left too little capacity to arrest everyone? Two officers laugh: “You might say that, we couldn’t possibly comment”.

The sun shines on them as Waterloo Bridge is magically transformed into the Boris Johnson £53m garden bridge that never was, with a long line of trees in big pots down the middle and rows of flowers. The calm of the carless river crossing is a good advertisement for how cities might be. “Come and join us, take time off work,” Gail Bradbrook, one of the organisers, exhorted Today programme listeners this morning. She’s right. All who can should be there. Those in tree-hugging outfits and carnival costumes may attract the cameras – but gathered here are all ages and occupations. The school strikes, started by the remarkable Greta Thunberg, caught the imagination around the world, just as this protest is echoed in 80 cities, from India to the US. Challenged by Nick Robinson for the nuisance caused, Bradbrook said “It gets you on the Today programme” – and that’s the blunt truth.

By another stroke of protesters’ luck, tonight David Attenborough delivers his most devastating warning yet in BBC One’s Climate Change: The Facts. In the past the BBC has been criticised for avoiding climate controversy in its phenomenally successful, world bestselling Attenborough nature franchise. No longer. Possibly spurred on by Netflix poaching Attenborough for its more polemical series, the gloves are off. “If we have not taken dramatic action within the next decade we could face irreversible damage to the natural world and the collapse of our societies,” he says. The 92-year-old naturalist speaks with resounding authority, as scientists lay out the undeniable evidence of what 1C extra heat has done already, and what risks becoming unstoppable. Among horrifying footage of wildfires, ice cliffs crumbling into the sea and permafrost melting to release methane many times more damaging than CO2, come eye-watering particulars: one foot of Louisiana is being lost every 45 minutes to rising sea levels. “It’s hard to exaggerate the peril we are in,” he says.

But as ever, how do we turn the Attenborough evidence and public protest into politics? We have been here before in 2006 when Gordon Brown brought over Al Gore to show his equally devastating film, An Inconvenient Truth. I thought then this might be a political tipping point. At least now the UK climate deniers are trounced: the BBC rightly refuses them airtime. But guiltily, I am well aware that every day we obsess in the media about other things is another day lost to the monumental, all-consuming threat to the planet and humanity. Greta Thunberg tells us to stop “talking about taxes or Brexit” and focus on the survival of life on earth.

It’s almost too big to think about. Or there is nothing new to say. The left/right divide has been a disaster: even if the respectable right no longer seriously denies climate change, it balks at solutions that look collectivist and socialist. Indeed, the need to take climate action, the need for state investment in a green new deal, gives new impetus to socialist thinking – an all-too-convenient excuse, the right suspects. Inequality blocks every remedy: how do you make everyone cut down on fossil fuels without helping the poor in cold homes and old cars, and giving aid to poorer countries in order to redistribute the pain? If the US, China, India or anywhere else isn’t doing enough, why should we? Reasons why politicians do too little are legion.

As Larry Elliot writes today, it helps that Mark Carney of the Bank of England is warning of catastrophic financial consequences. Will he follow that through in everything the bank does?

Extinction Rebellion wants net zero carbon emissions by 2025: existing targets are being badly missed. Protesters point to fracking, new coal mines, new incinerators and a Heathrow extension as perverse government polices. The government’s effective ban on land-based wind power just as it became profitable, to appease shire voters, was shocking. So was the sudden withdrawal of solar subsidies and the feed-in tariff, just as that industry was taking off, causing an immediate loss of 12,000 solar jobs. The government has asked the independent Committee on Climate Change to review its targets, results to be published shortly. What’s needed is a translation of fear into acceptable politics: the protesters are wise to demand a citizens’ assembly to examine all the evidence and gain public consent for what needs to be done outside the left-right battle-lines of conventional politics. The government should seize on that. Let’s hope the protest ends as elegantly as it has been conducted.

• Polly Toynbee is a Guardian columnist