This Egypt saw more action than IOTL since July 1941. Tobruq was besieged, Rommel was standing at Sidi Barrani. Now Hitler made the important decision: Barbarossa is delayed until 1942, he'll attack in the Med first.
Since the troops are already in the right place (in the Balcans), the Germans only had to transport them to the other side of the Med. At the same time, the attacks against Malta became stronger. Everywhere in the Eastern Med, the Luftwaffe attacks RN ships.
In August, the Axis troops in North Africa were strong enough to attack and take Tobruq. Now the way from Tripolis to the front was free. Rommel who's impatient attacked the Brits even before all of his troops have arrived. Still successful, the Brits were driven back to El Alamein.
In September, Rommel was ready for the final attack. He know had 10 divisions ready, plus some Italians, against 7+ (I guess Churchill would send them some, too, after they can guess what the Germans are up to - Max) of the Empire, plus a few others. Numerical superiority and Rommel's strategic talent were too much; before the battle became hopeless, they started to retreat. At the banks of the Nile, they were ready for the next stand. Fighting in the fertile river valleys and big cities was a completely new situation for Rommel (so I had to make things up). However, after a big battle end of September, Alexandria was taken. The Brits destroyed the bridges, but that was nothing that couldn't be repaired. Middle of October, Cairo and all of the Nile delta were in Axis hands. A few days later, the remaining British troops were behind the Suez canal.
Rommel's victory in the Nile delta made the situation for the empire difficult. Their troops in Africa and the Middle East now were disconnected from each other, the Germans controlled the important harbor of Alexandria, and without the Suez canal, the most important connection from Britain to India was cut. They managed to bring some support troops from India, but at the same time, the Nazis start to recruit Arabs willing to fight against the Brits, since they controlled Egypt's biggest cities. There's even unrest in Churchill's war cabinet. Some ministers and MPs around Halifax pointed out that Britain couldn't fight this war alone forever. Churchill resisted: “It's not over yet!”
Hitler now has to make a decision: Attack in the South, through the Nile valley, aiming for Sudan and Ethiopia, or East? Since he wanted the gulf oil, he quickly decided for the East. Rommel crossed the Suez canal and conquered the Sinai peninsula. The Italians and Arabs attacked in the South, but were stopped at the height of El Faiyum and Beni Suef. At the end of October, the Germans stood near the little city of Gaza.
In June 1943, the operation began. The attack of the Allies broke through the Axis' lines, throwing them back to Cairo (which they can't take at the moment due to lacking powers, since the nazis have organised a lot of Arabs willing to fight against the Brits - at least their propaganda says so), and is continued North-East, targeted at the Suez canal. This is the worst hit that Hitler got so far in the war, but he couldn't do anything now. Having control over the Red Sea, the Brits managed to land some troops at the Sinai, attacking the canal from two sides. Hitler had to defend the connection between his troops in Egypt and the Middle East respectively and ordered some troops from the Palestine front (where the Jewish freedom fighters still defended their pocket on the Med) down South, calculating that the Jews were too worn out for an attack. When they broke through the German lines between Jerusalem and Gaza and meet the Brits in the Negev, he learned that he was wrong.
In 1944, the Allies landed in North Africa. In May, Tripolis was conquered; in June, the Allies were in the Cyrenaica; and in July, the fight for the Egyptian delta had begun. At the same time, troops from everywhere from the Empire attacked the delta from the South. Now, unrest started to spread among Egyptian collaborators… they knew, if the nazis lose the war, they'll be in trouble, so now there'd be a good time to switch sides…
In October, the Allies were in Palestine and crossed the border to Lebanon and Syria; and in November, Generalfeldmarschall Erwin Rommel, cut off from any kind of support, had to capitulate with several armies.