Actually the fall of the British Empire is largely considered to mostly have taken place because of the Suez Crisis in 1956. Essentially, British Prime Minister Antony Eden and French Prime Minister Guy Mollet, with the help of Israeli Prime Minister David Ben-Gurion, attempted to depose the Egyptian government and take control of the Suez Canal. The United States refused to participate or endorse the action, fearing what the Soviet reaction might be.

The British-led invasion was incredibly unpopular and led to a ceasefire, followed by the resignation of Anthony Eden in 1957. His Chancellor of the Exchequer, Harold Macmillan became Prime Minister.

Macmillan was instrumental in propelling Britain into the 20th century, with rigorous reforms in many areas of government, chief among which was the massive decolonization of Sub-Saharan Africa. Under Macmillan all of the British colonies in Africa were either granted full independence, or at least had started the process by the end of his Prime Ministership in 1963.

Although it is true to say that WWII had a profound impact on Australia and New Zealand as Dominions of the British Empire. In 1931 Britain passed the Statue of Westminster which granted total autonomy to Canada and South Africa and offered it to Australia and New Zealand who did not accept right away.

After the Japanese attacked Pearl Harbour, but especially after the Fall of Singapore, the governments of Australia and New Zealand realized that Britain would be unable to fully defend them in case of a Japanese invasion. Australia enacted the Statute of Westminster in 1942 in response, and New Zealand followed suit in 1947.

It's also worth pointing out the complicated Indian reaction to WWII. Muhammed Ali Jinnah of the Muslim League encouraged Indian Muslims to sign up and fight so that they would have a better bargaining position for achieving independence later. Gandhi totally opposed Indians fighting in the war, and Jawaharlal Nehru initially tried to get assurances from the British that they would be granted independence after the war, which was not given to him, so he to opposed the war. Despite this, the Indian army was the largest volunteer force to fight in WWII.