Churchill moment seems to be fairly recent as a set phrase and doesn't even register in Google Ngrams or Trends. Google Insights for Search only has two small peaks (100 in May 2011, 71 in March 2012). I've not heard it as a set phrase either, although I can guess at its meaning.
It seems to become a set phrase towards the end of the 2000s, and used much more in the 2010s.
A question on the origin of the phrase is answered well in the Malaysia Star in January 2012, after it was heard in a December 2011 BBC podcast. Summarising the start, quoting the end:
The phrase “a __ moment” is believed to have originated with “Kodak Moment” first used by the Eastman Kodak Company in 1961 to advertise their photographic films and cameras. ...
Much later, in 1996, “a senior moment”, a variation of the above phrase, was used by Evelyn Weinstein who attended a conference on memory at Queen’s in New York City. ...
Other “moments” were then created by speakers of the language, like “eureka moment” and “blonde moment”. I hadn’t heard of “a Churchill moment” until I listened to your clip, but can try to work out what it means in the context of the discussion in the clip. ... The British statesman Winston Churchill did many things in his 90-odd years or so on earth. However, those who know about him would probably agree that his true greatness lay in what he did for Britain as its prime minister during World War II. He managed to prevent a Nazi German land invasion of Britain in 1940 by increasing the country’s preparedness for the Battle of Britain (an air battle) and the morale of the forces and the public. This he did through creative war planning (e.g. fast-track fighter plane manufacturing and fighter pilot training) and rousing rhetoric, both in parliament and radio broadcasts. In local terms he successfully promoted the “Britain Boleh” mentality at a most challenging time in the country’s history, and refused to concede defeat to or agree to an armistice with Nazi Germany.
“A Churchill moment” can therefore be interpreted as a creative and inspiring determination to overcome a difficult challenge. In the audio clip, the challenge is “the biggest challenge facing humanity: tackling climate change.” For this to be successful, Anderson says that people have to be weaned from using fossil fuels to using renewable energy, like solar energy, wind energy, biofuel, etc. He insists that this is possible if people have the will and the determination to do it, as Churchill had in WWII. And people must be persuaded to believe that “this is a Churchill moment” – we must act now, or lose the war against climate change.
In the 2000s, a Churchill moment is used in the news as a defining, decisive moment to lead the people forward. In each of the following, it is used to say "this is someone's Churchill moment" -- this is the moment they acted like Churchill, reminding us of his leadership during the second world war.
Bloomberg - Nov 25, 2008:
Brown's 'Churchill' Moment Masks Failure of Regulator He Built
Gordon Brown isn’t known for making quick decisions. He agonized for months over whether to call early elections after becoming U.K. prime minister in June 2007, before deciding against it.
... On Oct. 7, shares of HBOS Plc, Britain’s biggest mortgage lender, plunged 42 percent, while Royal Bank of Scotland Group Plc sank 39 percent.
Just a day later, Brown and Chancellor of the Exchequer Alistair Darling announced that the British government would spend up to 50 billion pounds ($75.6 billion) buying stakes in the country’s flagship banks to prevent a panic. ...
Brown was resurgent just weeks after lagging in opinion polls. French daily Le Figaro wrote that Brown’s calm, reasoned response in a time of crisis was worthy of Winston Churchill. Paul Krugman, the Nobel Prize-winning economist, pondered in his New York Times column: “Has Gordon Brown, the British prime minister, saved the world financial system?” Krugman said that would be an exaggeration, yet he praised Brown for his quick, clear-headed recapitalization.
Glasgow Sunday Herald - Sep 5, 2004:
BUSH: A MAN'S MAN IN the LAND OF THE FREE; ALL THE ...
What gets them most is the repeated image of Bush three days after the World Trade Centre attack, standing with his arm around a firefighter and a bullhorn in his hand shouting: "I can hear you, the whole world can hear you and pretty soon the people who knocked those buildings down will hear you." The Republicans say it is his Churchill moment - his "we will fight them on the beaches" speech. It is the moment when, to the Republicans at least, Bush became the true leader of the free world.
Sydney Morning Herald - Mar 31, 2003:
Dignity isn't all that Blair may be losing - War on Iraq -...
Tony Blair could still get his Churchill moment. Basra might fall, Baghdad could follow, with the British and Americans finally winning their long-promised tears-and-cheers welcome from grateful Iraqis. Blair would be vindicated as surely as Winston Churchill was 60 years ago.
Earlier news reports refer to Churchill's moment much more generally. A summary from from the Dallas Morning News - Nov 26, 1989:
Berlin Wall piece sought for...
Michael Vaughn, pastor of First Presbyterian Church in Fulton. "I would like to see the memorial here which notes Churchill's moment in history bracketed with the other end of history: the fall of the Berlin Wall."
Montreal Gazette - Nov 30, 1964:
For Sir Winston 10th Decade Begins
Another commemoration of Churchill's moment of destiny, timed for 90th birthday, is a tribute from Sir Isaiah Berlin called "Mr Churchill In 1940." In this slim book ... the Oxford scholar says Churchill's great gift was to mold the British People though his power of words so that they matched the "heroic mood" he created.