These are not quite the same kinds of cool. Cool authority was what district officers in imperial outposts exercised over the natives, a sangfroid as traditional as John Bull's waistcoat, and not what the BTA had in mind. It seems odd that the old cool, which stopped a fellow wincing when being beaten by prefects or shot at by tribesmen, should have come to be the word for a popular singer who sends his fans into raptures, substituting a bit of wild abandon, as it were, for stiff- lipped self-control. But you can see how it happened.
Cool jazz, a laid-back style of playing different from Dixieland or swing, became fashionable in about the 1940s, and it wasn't long before the idea that to be cool was to be fashionable grew into the idea that what was fashionable must be cool - a good example, I suppose, of the undistributed middle. So now anything that defies convention can be called cool. So, for that matter, can anything vaguely admirable. The more meaning it's given, the less it means.
But some of the older nuances are still there, like the one that means "cheeky" (youth putting up two fingers to age), in use long before the jazzmen nabbed the word. And in minimising the Union Jack the BTA is going back to a much misunderstood Kipling when he railed against "jelly-bellied flag-flappers", such as the visiting lecturer who so upset Stalky and his schoolmates by spouting emotionally about patriotism. That sort of patriotism was as unacceptable - or as we might say uncool - to Kipling as it is to most people now.