Assuming you are not doing other major POD's. The Allies still win. Let's keep the forces in theater near OTL levels, so we don't end up with the other topic "What if the Afrika Korp fights on the eastern front?"
So I will start with better Italian planning and staff work, but not something like stockpiling supplies in Libya assuming the French will fall to Hitler quickly and Italy planned for it. We have had recent threads on this topic. What can Italy do?
- Malta falls quickly once Italy joins the war.
- Adequate lead time can be given to the Italian generals on Greece. The Greek operations can go well.
- We can better supply, plan the operations in Italian East Africa. We can slow the UK down a good bit here.
- Attacking across the Alps does not work well. At least to me, there is an open question of what can Italy do in the few weeks after it is clear France will fall if there are good contingency plans. Is something like taking Tunisia, Corsica or Crete realistic?
- A well lead Italian forces can do most of what Rommel did, but with Italian forces plus maybe a little bit of German help. We have logistical issues here, so probably we end up with a better defended Sicily.
With all this, the Italians do better. I just don't see them holding Cairo for any length of time or much less taking the Suez. Maybe we can get the British Fleet to retreat to the Red Sea.
Now you said super lucky. Start with the base above, then lets look at really bad OTL performances. The defense of Singapore was a joke. The Japanese took with 1/3 of the troops. The USA in the PI allowed their air force to be largely destroyed on the ground. Or we can look at Japan at Midway. So what is super lucky. Italy attacks Tunisia before the fall of France. It works easily due to a jaw dropping set of mistakes by the French. British forces in the Med sortie to defend Tunisia, but things go badly and losses are really lopsided. Midway was 4:1. So make this 3:1 in capital ships, 2:1 in support ships. I guess super lucky might be Corsica working, since after all we are wanking the Italians. France makes peace. UK will still wipe out French fleet at some point, but French fleet gets some intel and this is actually a battle at sea where the UK loses some ships. Say trade capital ships 1:1. UK losses are enough that UK warships largely abandon central and eastern Med Sea. With better supplies, I guess the Italians/Germans can hold Cairo for a while and maybe have enough air power and naval forces to close Suez via mostly mines.
USA will likely still do Torch, but maybe 6 months later to use more forces. Better supplies, Maybe Tunisia can hold long enough against the USA to avoid any attacks on Italy. USA will still do D-Day about on time. Italy stays in war. War might last a few months longer since Germany will eventually move forces to slow Russians from Med area. I guess if USA puts a low enough priority on Africa, we might end war with stalemated lines somewhere in Algeria and Cairo falling in early 1945 to Allies.
Italians have really good military reputation ITTL, but not sure I have changed the larger war.