"Climate change has been a bit like everybody playing a deadly game of grandmother's footsteps for the last 20 years, now this climate change grandmother has turned around and started running towards us.
"It's touch and go whether we're going to survive what we've done."
Thompson has just returned from a trip to the Arctic where, she said, "the effects of the melting ice are written so clearly on the landscape".
She added that everyone has to act on cutting emissions.
But she said: "An international climate deal is of absolutely vital importance. It must be put into law. It can't be a non-binding agreement. Those agreements have fallen to pulp in our hands over the last 20 years."
She criticised David Cameron for encouraging oil, gas and coal, and said the politics of fighting climate change were "profound, and deep and dirty".
She added: "This is the battle of our lives. We're fighting for our children."
Greenpeace UK executive director John Sauven, who was also on the march, said scientists were clear about the dangers of climate change, and most world leaders understood the science, but were failing to respond to what needed to happen.
He said events were taking place today from Papua New Guinea and Australia to the march in New York.
"That is really what we need - global pressure coming from below on our political leaders."
He said Mr Cameron was going to the climate summit at a time when the UK was burning "unbelievable" amounts of coal, and called on the Prime Minister to announce a phase-out of coal power in the UK.
Leo Hickman, chief climate change adviser for conservation charity WWF-UK, said that while the risks of climate change had been well known for a decade or more, the opportunities associated with tackling the problem were emerging.
Recent research has shown that innovative technology and new investment in cities, energy and agriculture could cut emissions at the same time as saving money, boosting growth and improving health.
"We shouldn't be scared of decarbonising, it isn't going to wreck our economy. For the UK economy there's some really clear opportunities and co-benefits of decarbonising. We should grasp this moment rather than be scared of it," he said.
Campaign groups including Avaaz, Greenpeace UK, Christian Aid and Friends of the Earth took part in the march in London, while other events were taking place in cities around the UK.
Friends of the Earth's policy and campaigns director Craig Bennett said: "This is a global call for action that mustn't be ignored.
"It's time to end our dangerous addiction to dirty fossil fuels and plug into a clean energy system fuelled by the wind, waves and sun - with wealthy developed nations like the UK taking the lead.
"Half measures and empty promises are not enough. Our leaders must take urgent action to prevent the world speeding towards catastrophic climate change."
Christian Aid's senior climate change adviser, Mohamed Adow, who is taking part in the New York march, said: "People around the world have demonstrated just how much this issue means to them by coming out onto the streets of their cities in the biggest ever mobilisation for the climate.
He added: "There is still time for us to change our course and put the world on a trajectory which will secure a safe and healthy future for us all.
"But we need to see political commitments this year. We cannot leave it until next year's crunch meeting in Paris in December.
"The People's Climate March shows that the time for action is now. This week world leaders have a chance to prove they can be trusted to do the right thing."
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