Depended on the country. The Americans got out quick, but they had the advantage of having an agreement worked out before the war. The British got out quick, stopped for a bit, than resumed getting out quick. The Belgians basically tried to postpone Congolese independence for as long as possible, with mixed success-they lasted until 1960, and the country was indeed a complete mess without them. The Dutch, French, and Portuguese fought wars.

The Dutch didn't want to get out of Indonesia and fought a war to hold onto the place. It wasn't easy: the Indonesians fought far harder than expected, because prior to the occupation, Indonesian nationalism wasn't much of a problem. Although in many ways in the outer islands the Japanese occupation was far more brutal than the Dutch one, the Japanese encouraged Indonesian nationalism (most crucially, in the crowded home island of Java) during their rule of the archipelago, especially when it became clear that Japan would lose the war("If we can't have them, they aren't going to, either" summed up the feeling in Tokyo when it came to former European colonies by 1944). Around 1,000 Japanese officers and soldiers stayed behind and served as vitally needed military professionals for the Indonesians. The Indonesian War was very complicated due to the pent up communal hatreds and various Communist and Islamic seperatist movements: even today, the definition of an "Indonesian" is somewhat artificial. The Dutch managed to capture most major islands by 1949, but this was of little use when it came to world opinion-when Truman threatened to cut off Marshall Plan aid if they didn't recognize Indonesian Independence, the Dutch agreed to leave.

The French famously fought losing wars in Vietnam and Algeria, the latter of which was pretty nasty, even on Third World war standards, due to the fact that Algeria was considered part of France and had a very large population that was of European extraction, that often had lived there for over a century. The Algerian War was famously vicious, and in many ways was a repeat of the Dutch in Indonesia on a bigger scale. While France won tactically, they lost strategically and politically, not least due to world disgust at the brutality of their methods. Charles De Gaulle had coup attempts against him for deciding to recognize Algerian independence, and a million pied-noirs fled to France to avoid the wrath of the FLN-a wise decision. Many of the most assimilated Muslim Frenchmen today are of Algerian extraction, with grandparents who were pied-noirs or harkis.

The British, on a relative scale, prepared the best: after India wanted to leave the empire, it became clear that the rest would follow over the next decades to the reality based community in London. The Labour government engaged in a decolonization policy until 1951, when the Tories were swept into office. The British did put down colonial uprisings in Malaya and Kenya successfully (and by our standards, brutally, but not on the level of France in Algeria) during this period. However by 1957, Conservative Prime Minister Harold Macmillan saw the writing on the wall, as indicated by his famous "Wind of Change" speech in 1960. The early 1960s saw a spate of new states in Africa.

The Belgians, on the other hand, thought it was a good idea to not train the Congolese as much as possible, so that they could hold onto the Congo on an indefinite basis as they prepared for independence. Didn't work out well. The Congo became independent with 13 college graduates in the nation, and part of the reason the Congo is a mess today is because the Belgians deliberately tried to make it so.

The Portuguese were the last to give up their African empire, in the 70s. Portuguese dictator Antonio de Oleveira Salazar in general was a conservative fellow who had a good relationship with Rhodesia's Ian Smith. They only lost Goa due to war with India in 1962, and they kept Macau until the late 90s. This was due mainly to two differences between Portugal and other colonial powers:

1. Portugal was a right wing dictatorship until 1974. As far as right wing dictators went, Salazar was undeniably competent at home and was tactful about how far he went with repression-combine that with the Cold War, and Portugal was accepted in the NATO community in spite of the lack of democracy in the nation. But this meant that unlike most Western European governments, they didn't really have to care about the opinions of All Decent People. Throughout the Truman/Eisenhower era, there was a tacit "understanding" with the US about the colony issue, unlike with France or the Netherlands. Some historians have mentioned that President Kennedy advised Salazar to grant independence. I don't know much about his relationship with Salazar, but I find it plausible given Kennedy's general relationship with older European statesman such as Adenauer or De Gaulle. However, if true, this had no absolutely effect whatsoever on Salazar.

2. Portuguese colonial policy had traditionally stressed "intermixing" and "becoming Portuguese", unlike most colonial powers, perhaps because they had been at it for far longer than any other colonial power. Salazar saw the colonies as a way of making sure Portugal remained relevant. Moreover, he and many Estado Novo figures also saw the whole empire as literally part of Portugal, like France did with Algeria, but not with most French colonies. Salazar in 1962 argued that Goans were not Indians, but Portuguese of Goa, and this had more of a kernel of truth to it than it did with every other colonial power: Goans were found in the cabinet and enjoyed full rights as Portuguese citizens. To this day, Goa is very different from the rest of India, even by the standards of the diverse country-Catholicism is a major religion, wine and pork are commonly found in the food, etc. Because of this, Lisbon was a die-hard when it came to the Wind of Change. However, the Indians won Goa in 1962, and by this time, war had also broken out in Angola. Salazar and his successor, Caetano, nevertheless refused to budge throughout the 60s and early 70s, and Portugal's NATO colleagues were dealing with enough problems to not force the issue. It was only when the left-wing Carnation Revolution occurred in Lisbon, which led to democracy in Portugal, that Portuguese withdrawal occurred.

America had already agreed to grant independence eventually before WWII to the Philippines, so there was no independence struggle, and within a year or two of the war ending, Manila had full independence.