Most recognize that moment as being the Suez Crisis of 1956. Anglo-French forces invaded Egypt with the aid of Israel, to take control of the Canal back from Nasser's Egypt. The Egyptian air force was destroyed, and the army badly beaten, but the Soviets threatened nuclear attack and Ike put pressure on Eden to pull out. It also sparked considerable opposition from the British public, and marked essentially the last time that Britain OR France would try to act as a Great Power on the World stage, without American support.

As DuxBelisarius pointed out, it was the Suez Crisis in '56.

As to how Britain reacted? Well shortly after the Crisis British PM Anthony Eden resigned and his Chancellor of the Exchequer, Harold Macmillan, took over.

Macmillan, Supermac as he was popularly known, set about the process of ending empire. Under his stewardship almost all African colonies were granted independence and Britain moved to distance itself from the image of the old Imperial power.

Looking forward Macmillan thought that Britain could be a king maker in Europe, and so set about applying to join the various European communities that were the precursor to the modern EU.

As an interesting sidenote, almost the exact same thing happened to France, though slightly earlier. Following the expulsion of Japan from Vietnam, France tried to retake it. Look up the Battle of Dien Bien Phu to see how that went (Hint: Not very well). Charles de Gaulle, as well as other French Presidents, decided France no longer could be a world power, but it could be a power in Europe. This led to the founding of the European Coal and Steel Community which is sort of the foundation the EU is built on.

I bring this up because every time Britain applied to join the ECSC or the EEC Charles de Gaulle vetoed them out. It took Britain about 20 years to actually succeed in joining the EEC because France wanted to be the European power so blocked Britain at every turn.