I think they could do this...
Greece and Crete
The Germans "Make Faces" at the British home defences from June to August 1940. This would start with the an anti-shipping campaign that the Luftwaffe conducted around the British coasts. However, instead of trying to destroy Fighter Command in August they would go straight into the Blitz.
Meanwhile they were making preparations to invade Greece from Bulgaria in October 1940. This was followed by an invasion of Crete in November 1940. Both operations might be less costly than the real ones for the Axis because the Greeks would be less prepared and the British would have less time to send help.
IOTL the Germans used Crete as an air base to:
-Attack the convoys from Alexandria to Tobruk and Malta. The sea between Crete and Cyrenaica was known as "Bomb Alley;"
-A staging post for transport aircraft carrying supplies to Cyrenaica;
-Attack Alexandria and the Suez Canal.
ITTL they would be able to do begin the above 6 months earlier.
Yugoslavia
ITTL the Germans don't bring forward the invasion of Yugoslavia with the invasion of Greece. The Yugoslav Government keeps the country neutral until Soviet forces overrun Romania at which point it declares war on Germany to avoid the same fate. However, the Soviets invaded Bulgaria IOTL when it wasn't officially at war with the USSR, so they might invade Yugoslavia anyway ITTL.
In the meantime it might be easier for the Germans to obtain raw materials from Yugoslavia than was the case IOTL. It releases ground troops for other fronts because no occupation force is required from 1941 to 1944. IOTL USAAF bombers used bases in southern Italy to attack the Rumanian oilfields and aircraft factories that the Germans had built in Hungary. But to do that they had to fly over Yugoslavia. They might refrain from doing that for as long as Yugoslavia remains neutral.
The Eastern Front
Bringing forward the invasion of Greece by 6 months and not invading Yugoslavia at all releases resources that were used to set up the bases that the invasions were launched from IOTL for other uses. One possible use is to expand the bases that Operation Barbarossa was launched from. But, as this is a Mediterranean thread and not an Eastern Front thread the Germans use the resources to build up the Luftwaffe's ground organisation in Crete, Sardinia, Sicily and Libya.
(Take cover! Incoming from logisticians about the impracticality of reinforcing Libya!)
However, if the Balkan Campaign of April to May 1941 did delay Operation Barbarossa for 6 weeks IOTL, the delay is butterflied away ITTL because the Balkan Campaign happens 6 months earlier.
Malta
In the summer of 1939 the air defence requirements of Malta were assessed as 48 or 64 fighters in 4 squadrons (depending on the number per squadron), 112 heavy AA guns, 60 light AA guns and 24 searchlights.
A year later it actually had 3 Gloster Gladiator fighters, plus another Gladiator in reserve, 34 HAAG, 12 LAAG and 24 searchlights. The number of British infantry battalions had been increased from 4 to 5 by transferring one from Gibraltar, but the requirement was for 7 plus the King's Own Malta Regiment.
IOTL the air defences of Malta had by the beginning of 1941 been upgraded to one fighter squadron (16 Hurricanes, plus another 4 Hurricanes in reserve), 74 HAAG, 34 LAAG and 24 searchlights. The number of infantry battalions had increased to 6 plus the KOMR. IOTL the British had to send RAF squadrons to Greece along with army troops to guard the airfields and lines-of-communication, but the earlier conquest of Greece and Crete by the Axis butterflies that out. On the other hand that is more than cancelled out by the extra forces the Germans send to the Mediterranean ITTL and that means that Malta might not receive any of the reinforcements that it received IOTL.
ITTL the German strategy was to mount an airborne invasion of Crete in the late autumn of 1940 followed up by an invasion of Malta in the spring of 1941 to secure the southern flank before the invasion of the USSR which was scheduled for the summer of 1941. Meanwhile Malta would be isolated and softened up by the Luftwaffe deployed in fliegerkorps strength in Crete and another fliegerkorps operating from Sardinia and Sicily.
Work on building up the necessary ground organisation in Sardinia and Sicily began in July 1940 using the resources that IOTL were used to build up the ground organisation in France for the Battle of Britain. Operations against Malta began in October 1940, which was 3 months before they started IOTL.
Operations from Crete would begin considerably later. The island would not be captured until November and it would then take time to build up the ground organisation. However, the Germans will be able to turn it into an effective base 6 months earlier than they did IOTL. In the mean time the earlier invasion of Crete might make more intensive air operations from Italian bases in the Dodecanese feasible and they are a bit closer to Egypt. Also the aircraft that belonged to the CAI that operated against Britain in late 1940 IOTL were instead used to reinforce the Regia Aeronautica in the Mediterranean.
ITTL Malta receives fewer reinforcements than it did IOTL rather than more. The Luftwaffe attacks the supply convoys from October 1940 and the losses were high enough for the British High Command decides to suspend further shipments because the passage to Malta has become too dangerous. Furthermore the earlier Axis occupation of Crete might make the British High Command think that all available forces have to be concentrated on the defence of Egypt.
Therefore an invasion of Malta in May 1941 in place of the invasion of Crete IOTL might not be necessary because the island might have been starved into surrender by then.
Libya
The Germans do not send any front-line ground and air forces to Libya any earlier than they do IOTL. However, they do send technicians and equipment to help the Italians increase the capacity of Libya's ports and inland transport network. But any improvement this brings about does not come into effect until the second half of 1941.
The German Government puts pressure on Vichy to let the Italians use the Tunisian port of Bizerta, which Mussolini asked Hitler to do IOTL. From there they go by rail to Tripoli (note that I have cheated and had the railways built to the metre gauge instead of 950mm to match Tunisia's and extended the line that ran west from Tripoli to the southern terminus of the Tunisian railway system.) Then as IOTL the supplies proceed by road to the Egyptian frontier.
The use of the Tunisian ports does not mean that any more supplies reach the Italian troops at the other end of Libya. However, it does allow the more efficient use of the available merchant shipping which would be required to supply the Luftwaffe in Crete, Sardinia and Sicily. There is no reduction in Axis shipping losses on the Libyan supply routes until the middle of 1941 ITTL because they were negligible to then IOTL also. But after that there is a reduction because they were using a safer route or Malta had surrendered.
The earlier invasion of Greece led to the cancellation of Operation Compass and the Italians don't loose Cyrenaica. That butterflies away the loss of most of the Italian army in Libya. The Axis use the troops weren't lost in Operation Compass as labour to extend the railways that radiated (not very far) from Benghazi IOTL and as labour to unload the supply ships.
However, it also means that the British don't loose the equipment of 3 infantry divisions and one armoured brigade in April-May 1941 ITTL and there is no Siege of Tobruk.
The Afrika Korps and a Luftwaffe contingent is still sent to Libya to bolster the Italians. As there is no Tobruk to relieve, the 8th Army probably doesn't attempt Operations Brevity and Battleaxe so it can build up its forces for Operation Crusader.
The result of Operation Crusader depends on how effective the Axis attempts to improve the capacity of Libya's ports and inland transport system have been. I think it would be a draw. Therefore the front line would remain on the Egyptian frontier and the Axis don't loose the troops that became prisoners when the British took Cyrenaica for the second time.
Therefore the Axis forces are that bit closer to Alexandria at the start of their May 1942 offensive. Furthermore the capacity of their supply lines would have been increased. However, the British would strengthen their defences at Mersa Maruth and complete the Alamein Line instead of building the Gazala Line
Relative Attrition
No Battle of Britain greatly reduces Luftwaffe losses between July and the end of September 1940. The aircraft and aircrew not lost IOTL provide the air forces required for the October 1940 offensive in the Mediterranean ITTL. Some of the aircrew killed or captured in the Battle of Britain IOTL will suffer the same fate in the Mediterranean before May 1941 ITTL. However, I think the losses sustained ITTL would be significantly lighter than the losses of the real world.
As a result the Luftwaffe would have more aircraft and a larger pool of trained aircrew at the beginning of Operation Barbarossa. Therefore the Luftwaffe did not need to raid its training organisation of instructors and aircraft to bring its front-line up to strength. This might result in the quality of the Luftwaffe's aircrew declining at a slower rate as the war progressed than it did IOTL.
No Battle of Britain also means the RAF suffers lighter losses before October 1940. However, that does not mean they can automatically send 20 squadrons to reinforce Egypt in the last quarter of 1940.
Firstly the Amphibious Mammal that cannot be named is only ASB as long as the British maintain strong enough home defences to make it ASB. This means that priority has to be given to building up a strong Fighter Command. Although the Germans start a major air offensive in the Mediterranean ITTL in October 1940 the British High Command might think that this is a diversion for an invasion of Britain in the spring of 1941. This is because the Germans have the advantage of interior lines of communication and therefore can switch aircraft between fronts faster than the RAF can. Therefore the.
Secondly the Blitz still happens and the British have to maintain an effective day fighter force to deter the Germans from attacking by day as well as by night.
Thirdly even if the RAF can spare more fighters for the Mediterranean Front it will be at least January 1941 before they become operational. The aircraft can be flown to Egypt via British West Africa and The Sudan. However, the ground crews and their equipment will have to go by sea around the Cape of Good Hope. Meanwhile the air bases they will operate from have to be built.
