Especially in the Middle East, the local people were hostile to the Colonial Powers like Britain. We can imagine that the Iraqi Rashid Ali was encouraged by the British difficulties in Greece and North Africa to harass the British in Iraq. Rashid Ali was influenced by the pro-German Grand Mufti of Jerusalem. A coup d'état overthrew the pro-British regent and installed Rashid Ali as Prime Minister of Iraq. Many of the same themes as in present times were present in 1941. There was the strong Arab nationalist and Islamic fervor that exists now.
The British responded to the coup by sending the 20th Indian Infantry Brigade to Basra, the oil terminal. The next move by the Iraqi government was to send troops to the RAF base at Habbiniyah on the Euphrates. The base was just 50 miles west of Baghdad. The RAF mostly had obsolete aircraft at Habbiniyah that were used for training. They were enough, however, to establish air superiority over the Iraqi air force. Once enough British and Indian Army troops had been sent to Iraq, they were able to overcome the Iraqi Army and lift the siege and capture Falluja and Baghdad. While the action to remove Iraq as a factor continued, it was at an inconvenient time, opposite the end of Greece and the Crete campaign, as well as the fighting in Libya, on the western desert in Cyrenaica. There was also the Tiger Convoy and the need to resupply Malta all happening at the same time.