I can't speak for the UK, but Americans learn nothing about the nonaligned movement or really anything about how everybody that wasn't the USA or USSR dealt with the USA and USSR.

De Gaulle had little-to-no trust of the British or Americans because he viewed the Americans as opportunists and the British as island-dwellers whose involvement on the Continent was 100% defensive. Britain didn't send troops to fight Hitler until the Nazis invaded Denmark; the US didn't declare war on Germany until after Germany declared war on the US because the US declared war on Japan (which was a member of the Tripartite Pact with Germany and Italy). Europe has always been "over there" to the Americans and the "Phoney War" to the UK while to the French it's the Maginot Line.

De Gaulle was the Minister of Defense during the Suez Crisis, in which the USA pinned back the ears of the UK and France for fucking about in the Middle East and giving the Soviet Union cover to crush Hungary. From a European perspective, that's the United States and its puppet the United Nations dictating what Europeans are allowed to do in their own interests.

As an American, it's really easy to see that the world is a better place when America calls the shots. As anyone non-American can tell you, though, you step outside of the funhouse for one minute and empires look like empires.