I am going to ignore the 1940-1943 fighting in North Africa. It is to complicated to write about here. Erwin Rommel and Bernard Montgomery are very famous figures in World War II, so the fighting in the Western Desert of Egypt is fairly well known.
East of the Suez canal, World War II is far more obscure, but there was some fighting going on. Shortly before Great Britain declared war on Germany, the Mufti of Jerusalem fled Palestine and went to Baghdad. He was a fanatical anti-Zionist and was able to convince the Iraqi people to elect Rashid Ali as the prime minister of Iraq. They formed a conspiracy with three other prominent officers of the Iraqi army. A huge mob began to protest in front of the British embassy in Baghdad. By the end of March 1941, Emir Abdul Ilah, the pro-British Regent, fled from Baghdad. This concerned the British so much, they redirected the 10 Indian Army division, which was sailing to Malaysia, to go to Basra instead. Rashid Ali was informed of this move, but he denied the British permission to send any more troops to Iraq. The Iraqi army then decided to attack the RAF training base at Habbaniya (80 kilometers north west of Baghdad) This remote desert training base had its population swell to 9000 British people, including women and children, that had fled the turmoil in Baghdad, but the 265 soldiers that were guarding the base were able to hold off the initial attack. They were helped by the 88 RAF Gladiator fighter planes, but the 10th Indian army division were able to chase the Iraqi army away by April 29 1941 and topple Rashid Ali as the prime minister of Iraq. Iraq was nominally independent, but signed a treaty with Great Britain that made it a protectorate. Iraq elected a pro-Axis government in late 1940, but its revolt against the British Empire was short lived and quickly crushed.
Rashid Ali was getting support from the Italians and Germans, via the Vichy French controlled colony of Syria. In June 1941, Winston Churchill and the British War cabinet ordered General Wavell, the British Army commander in the Middle East, to invade Syria. Wavell was getting ready to launch Operation Battle Ax, an attack on the Italians and German Africa Corps, while dealing with the German invasion of Crete and finishing up the destruction of the Italian East African empire in Ethiopia. Wavell was able to scrape together two thirds of the Australian 7th infantry division, one third of the British 6th infantry division and two thirds of the 1st Cavalry division to attack Syria. When the Vichy French and the thirty thousand troops they had defending Damascus saw how weak this attack was, they put up a stiff defense. Wavell had to send in the 10th Indian division, five battalions of Free French troops, plus some New Zealand soldiers, that were evacuated from Crete, to bolster this attack. Damascus fell on June 21 1941. Beirut surrendered four weeks latter. This closed the door on any additional meddling by Axis agents in the Middle East.
In order to consolidate their position even further, Great Britain and the Soviet Union both invaded Iran. They eliminated the shah who was showing some pro-axis leanings. The United States did not participate in the initial invasion of Iran, it occurred before Pearl Harbor was attacked, but they ended up sending a larger garrison to Iran than either the British or the Soviet Union. The Americans built a better port in southwestern Iran and rebuilt the railroad between that port and the border with the Soviet Union. This was one of the main routes used to deliver Lend Lease goods to Russia.
Source: "The Grand Alliance" Volume three of Winston Churchill's six volume history of World War II.
Despite his masterful juggling act, fighting battles in Greece, Libya, Ethiopia, Iraq and Syria with limited forces, after the failure of Operation Battle Ax, Winston Churchill replaced General Wavell with General Claude Auchinleck.