Used to describe the contemporary culture of the United Kingdom, primarily during the 1990s.
Cool Britannia was little more than a mercifully short-lived piece of media and marketing hype aimed at promoting the UK to a world audience. It was coined in that context during the 1990s to exploit the popularity of various 'Brit Pop' bands, for example Oasis, Blur and Pulp and of The Spice Girls and the contemporary notoriety of 'Brit Art'. It was encouraged, some would say engineered by, the spin doctors of Tony Blair's 'New Labour' government. Various people who were thought to portray the image the government was keen to promote were invited to 10 Downing Street to be photographed shaking hands with the Prime Minister. Few of them have held on to any long term endorsement of the party though. Noel Gallagher, who was a high-profile visitor to No. 10 said later, in his usual style - "If I could be arsed enough to vote now it would be for the Liberal Democrats."
Obviously 'Cool Britannia' alluded to the song 'Rule Britannia'. That playing on words with the song is also to be seen in the anarchist slogan 'Britannia waives the rules'.
The term had been used before. It was the title of the opening song on the Bonzo Dog Doo-Dah Band's inspired 1967 album Gorilla. This little snippet was sung to the tune of Rule Britannia:
Britannia, you are cool
Take a trip!
Britons ever ever ever
Shall be hip
That comes from the 'Summer of Love', the era of a precursor phrase - 'Swinging London', when Britain, or London at least, really was cool.
It is unlikely that Cosmo Landesman was thinking of the Bonzos when he used the term in an article in The Sunday Times in August 1993, about a performance by U2:
"The children of cool Britannia may not know much about trigonometry, but they do know every art term in the book. On the other hand, your average American rock fan probably thinks dada is somebody you can borrow the car from."
It didn't catch on then and he had another go in the same paper in November the following year:
"... cool Britannia rules all the new waves."
The first example I can find from the 1990s where the phrase is made into a proper name by giving capital letters is in a piece by Waldemar Januszczak in December 1995, also in The Sunday Times, titled 'Cool Britannia'. That discussed the rising popularity of British art in the USA.
By 1998 the term had been seen for what it was and The Economist published an article, also titled 'Cool Britannia', that said what many people had already decided for themselves:
"It started life in 1967 as the title of a song by the "Bonzo Dog Doo Dah Band". By 1996 it had become a new flavour of Ben and Jerry's ice cream and the title on a Newsweek cover. Now it is government policy. Many people are already sick of the phrase 'Cool Britannia'."