The British occupation of Egypt was really something of a fluke. While Britain wanted access to the Suez Canal they also had stated goals of withdrawing troops from Egypt before WW 2 began on more than one occasion. In this what-if, the British as they proposed, withdraw from Egypt in 1922 and it becomes an independent nation. If this had occurred, Britain would have no military presence in that nation in 1940 when Italy enters the war. Would Italy invade? Would the Egyptians side with the Axis? The Suez Canal really makes no difference at that point since: A. The RN would not be based in Egypt or Alexandrea, but more likely in Aden or South Africa. B. Convoys couldn't be run through the Med in any case so its use for shortening movement to India and the Far East would be much more problematic. For the Italians control of the canal in either neutral or friendly hands means they have access to their base at Massawa and Italian East Africa for reinforcement. The Italian fleet could, theoretically, operate in the IO and Persian Gulf in this scenario. Could the British have managed to get a force together to take out Italian possessions in the time they historically did it in this scenario? I doubt it. On the other hand, it is likely Singapore would have had much better units defending it and that Japan might well have lost that campaign. Malta would have been almost certainly doomed. Greece, Cyprus, and Crete would not be economically feasible to support by Commonwealth troops. Given the stalemate in Greece, without British intervention, the Germans might not invade Yugoslavia and Greece as they did.