In OTL, after having conquered Yugoslavia and Greece, Hitler still decided to invade the Soviet Union on June 22nd of 1941.
But WI he had decided to advance further around the Mediterranean - striking against Malta, Egypt and even further?
Since the troops are already in the right place (in the Balkans), they only have to transport them to the other side of the Mediterranean. At the same time, the attacks against Malta become stronger. Everywhere in the Eastern Mediterranean, the Luftwaffe attacks RN ships.
In August, the Axis troops in North Africa are strong enough to attack and take Tobruq. Now the way from Tripolis to the front is free. An impatient Rommel attacks the Brits even before all of his troops have arrived. Still successful, the Brits are driven back to El Alamein.
In September, Rommel is ready for the final attack. He now has ten 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) of the Empire, plus a few others. Numerical superiority and Rommel's strategic talent are too much, before the battle becomes hopeless, they start to retreat. At the banks of the Nile, they are ready for the next stand. Fighting in the fertile river valleys and big cities is a completely new situation for Rommel. Still, after a big battle end of September, Alexandria is taken. The Brits destroy the bridges, but that's nothing that can't be repaired. Middle of October, Cairo and all of the Nile delta are in Axis hands. A few days later, the remaining British troops are 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 are disconnected from each other, the Germans control the important harbor of Alexandria, and without the Suez canal, the most important connection from Britain to India is cut. They managed to bring some support troops from India, but at the same time, since they control Egypt's biggest cities, the Nazis start to recruit Arabs willing to fight against the Brits. There's even unrest in Churchill's war cabinet. Some ministers and MPs around Halifax point out that Britain can't fight this war alone forever. Churchill resists: "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 wants the gulf oil, he quickly decides for the East. Rommel crosses the Suez canal and conquers the Sinai peninsula. The Italians and Arabs attack in the South, but are stopped at the height of El Faiyum and Beni Suef. At the end of October, the Germans stand near the little city of Gaza. Now, Churchill exchanges several messages with Jewish leaders like Ben Gurion. The Jews now may officially form an army to fight against the Germans. Since he mustn't anger the Arabs too much, he makes Ben Gurion promise that Arabs have to keep full rights in case a state of Israel should be founded after the war. As fast as possible, the Jews raise troops. They don't exactly have the best arms, but are eager to fight against the Nazis. Southern Palestine (the Negev desert) is conquered by Germany, but then the Brits and Jews manage to stop them somehow, often simply by bluffing (did you know that empty gas canisters pulled over the ground sound much like moving tanks? I'm not kidding, the Israelis invented that trick during their war of independence). News spread fast: Though Roosevelt doesn't declare war on Germany, he supports the Jewish brigades where it's possible. Many Jews in America (emigrates and natives) volunteer to fight in Palestine, others collect more money than ever to buy arms and transport them. Even non-Jews join their rows (many of them are veterans of the Spanish civil war or other socialists).
Middle of November, Hitler is getting furious since Rommel didn't break the lines yet. He demands a victory, or else. Rommel manages to find a way: Instead of attacking Palestine directly, he lets his troops march into Transjordania. One week later, Amman is conquered. It's easy to see what Rommel is up to: He wants to enclose the Jews and Brits in a pocket and forces them to fight on two fronts. They still manage to hold the line along the Jordan and the heights of Golan. End of November, Rommel wins a battle against Free French troops and approaches Damascus. About that time, anti-British rebellions start in Iraq (again). Hitler thinks that the time has come and orders to take Cyprus.
In the first week of December, things look bleak for Britain. Half of Cyprus and most of Syria are in German hands. On December 6th, they reach the coast of Lebanon. Palestine is cut off, while the Germans in Syria approach the rebels in Iraq. Again, Churchill exchanges messages with Ben Gurion, asking whether he wants to evacuate his troops from Palestine. Ben Gurion answers: "We're ready to die for Israel!" Churchill tells him: "I hope you know that we can't support you forever." Ben Gurion: "As long as our volunteers will run the German lines, we'll stand and fight." What they don't know: Hitler just ordered Rommel not to attack the Palestine pocket but only to clean up Syria and Mesopotamia. "We've got the harbors of Syria, now we need the oil!" he commands. He also orders that some Luftwaffe divisions are to be stationed in the North of Syria and Iraq - for an attack against the oil fields of Baku.
End of 1941 the Italians in Ethiopia capitulate, not much later than OTL. Now Churchill finally has a victory, which does a lot for British morale.
Then, everything changes with Pearl Harbor happening...
To be continued...