I only know sufficently about the Dutch East Indies to give an in depth answer on that. However I do know that French Indochina (consisting of what is today Vietnam, Cambodia and Laos) was ceded to the Japanese by the Vichy French government. Initially, the Vichy French only surrendered Hanoi and Saigon to the Japanese, but they eventually expanded their control over the rest of French Indochina. This actually led to the USA implementing steel and oil embargoes on the Japanese, which many would argue that this was one of the factors that led the Japanese to launch their attack on Pearl Harbor and the rest of South East Asia, including the Dutch East Indies, which had a notable amount of oil production facillities, among others.

While the Netherlands itself fell to the Germans in May 1940, the Dutch East Indies did not 'transfer' over to the Axis powers. The Dutch government in exile actually declared war on Japan on 8 December 1941, and fought the Japanese together with the Australian, American and British, under the ABDA Command structure. Dutch army and navy units were involved in the fighting over the Dutch East Indies but were defeated by the Japanese. Many of the troops ended up being POWs and scattered throughout the Japanese Empire in various prison camps.

While Indonesian nationalism was in its infancy in the early 20th century, the Dutch authorities often clamped down on such movements by arresting the leaders of such movements and playing on regional differences. In fact, future President Sukarno was arrested thrice by the Dutch authorities, and only freed in 1942 by the Japanese. Unlike the Dutch, the Japanese actually encouraged the burgeoning nationalist movements of their captured territories in South East Asia. Indonesia was no exception. One of the most cruical differences between the Dutch and Japanese administrations were that the Japanese allowed locals to be in much higher positions of power as compared to the previous colonial administration, where the highest echelons of leadership were reserved for the westerners. Another factor was that the Japanese decided to arm and train the local Indonesians for fear of Allied Invasion. They created the Pembela Tanah Air (PETA) in 1943. While they were supposed to disarm after the Japanese Surrender in August 1945, many leaders in PETA went on to join the revolutionary cause post WWII. Thirdly, the defeat that the Japanese inflicted on the Dutch destroyed the myth of invincibility that the West had on their Asian colonies, and the protection of their colonies from foreign invasion was a crucial building block of a stable colonial rule.

While the defeat of the Netherlands in Europe did not lead to her colonies falling, their defeat in the Dutch East Indies caused them to be unable to clamp down on the rising sense of nationalism among its people. Taking into consideration the Japanese policies, it caused a much strengthened nationalist movement by the end of the Second World War. In fact, Sukarno and Hatta issued the declaration of independence on thr 17th of August 1945, just two days after the official Japanese surrender. The cause recieved such popular support that when the British arrived at the city of Surabaya to take over as representatives of the Allied Powers, they were engaged by the local permuda in the Battle of Surabaya. This was to be the beginning of a four year struggle for independence which eventually led to the independence of Indonesia in 1949.

This is only a summary of what actually happened as while I can elaborate more on the rise of nationalism in the Dutch East Indies, it is not exactly relevant to the question