Well, Iraq would be the go-to example probably. Not a "colony", more of a dubiously independent dominion-country-thingy, but an interesting story nonetheless.

Iraq had come under British rule after taking it from the Ottomans in World War 1. Ruled as a League of Nations mandate, Iraq was nominally independent by 1932, but just like with Egypt, the UK had secured for itself military access to and military bases inside Iraq through a bilateral treaty. The motive was not as much the strategic importance of Iraq's geographic position, but the strategic importance of the vast oil reserves of the country - in fact, the British made sure to keep Iraq and Kuwait separate in order to limit the access to oil in either country. This lack of military importance is further exemplified by the fact that the UK did not actually use its right to position troops besides small air and naval formations in Iraq/Kuwait by the 1930s - which to Iraq sent the signal that the British were not all that concerned with Iraqi allegiance and that they could do their own thing.

In 1939, Iraq initially went along with the Allied war effort in World War 2. Although staying short of declaring war on Germany, they broke off the (already limited) diplomatic relations they had with Berlin.

That is until Rashid Ali al-Gaylani enters the picture. Prime Minister from early 1940 onwards, secretly reopened diplomatic channels, particularly with the German ambassador in Ankara, Turkey. He also did not bother to also break off relations with Italy like Iraq had done with Germany when Mussolini entered the war in summer of 1940. Italy's well-established diplomatic center in Baghdad became an operational center for open and disguised Axis activity, rallying an anti-British narrative.

Rashid Ali in April of 1941 launched a coup against the Prince Regent of Iraq, Abd al-Ilah. The prince fled to British hands and Rashid Ali formed the "Golden Square" with three other officers to seize power. The military government did not establish a republic, but replaced the regent with a puppet. Rashid Ali was once more put into office as Prime Minister and subsequently tried to secure German military assistance in mid April and reinforced Iraqi relations with Italy. The Golden Square persecuted pro-British politicians, driving several into exile.

Britain quickly determined that the strategic value of Iraqi oil shipped through neutral Turkey and into German-aligned Bulgaria and Romania could be the lifeline winning the war for the Axis Powers and Churchill decided in early April to militarily invade as to prevent Iraq from becoming an asset in the Axis war effort. The British realized that Iraq would never dare to join the war and would prefer neutrality, and they also realized that Turkey would not allow to become a tool in Iraqi isolation from international trade, so invasion was deemed the only way to keep Iraqi oil out of German tanks. The British armed forces gathered troops from all over the colonial empire, particularly India, and unloaded them in British Kuwait as well as Iraqi Basrah. This was still legal as per the treaty between the two nations signed prior to Iraqi independence, so Iraq realized that it technically had not been illegally invaded and moved to defensive positions on 30th of April, anticipating the impending British attack. The Iraqis tried to cut off the British air bases deep in Iraqi territory, particularly Habbaniya air base, where the war also started on the 2 May 1941. The Royal Air Force was the British main asset against a numerically powerful Iraqi ground force which had laid siege to Habbaniya. British relief forces heading in from Palestine routed the Iraqis and took the offensive.

On 11 May, Luftwaffe units began to arrive, redeployed in a hurry by Germany to aid Iraq when it became clear that Britain would take the military route. The German involvement actually led to the Soviet Union taking a diplomatic stance in favor of the Golden Square just short of a month before Germany stabbed the USSR in the back - the two were at this point still "friendly" "allies". Vichy France, Germany's French puppet, also provided valuable air bases to the German operations in Syria and Lebanon. On the 27th, Italian air units joined their German comrades and were integrated into the German air command.

But it was already to late, as on the same day, British ground forces had reached Baghdad - the universal sign of Iraqi defeat. The Golden Square government's position had become untenable and they fled to Iran and then to Germany.

Iraq and the UK agreed on an armistice on 31 May 1941. The regent returned on 1 June and began forming another pro-British government. Britain had succeeded in conceding how numerically outnumbered their forces truly had been by Iraqi land units and the Iraqis stood down, pacified and deterred once and for all from any pro-Nazi diplomatic experiments.

The remaining German and Italian air units had retreated to Syria and then to Europe in the evening of the 28th.

Related question: how much impact Azad Hind and their Japanese support have on post war Indian independence movement?