The Syrian campaign started prior to the German invasion of Russia. Prior to that, Germans and Italians were only fighting in the Western Desert against the British. The British seem to have attributed exaggerated prowess to the Germans, as the British expected that Russia would quickly collapse under German attack. Russia certainly had a dysfunctional government, but did have a better army than the western countries realized. British thinking was that Germany would take Cyprus and Syria after winning the battle for Crete. They imagined that they might see Germans advancing into the Middle East from the Caucasus.
The British were stronger during June to August than they had been. With the German air force diverted to Russia from the Mediterranean, the British were able to add air strength to Malta with the idea that they would be able to operate against the supply convoys to Libya from Italy. General Auchinleck took over as theater commander from General Wavell, who went to India. The army received new units and equipment. They had new aircraft from the United States and new tanks from there as well. They were in the process of forming the 10th Armoured Division and had a the beginnings of the 50th Infantry Division. Forces from East Africa were now available to the Western Desert after the defeat of Italian forces in Abyssinia. African troops were left to finish off the remaining Italian forces in East Africa. The 1st South African Division, and the 4th and 5th Indian Divisions were sent to the Western Desert.
The British sent Sir Oliver Lyttleton to the Middle East to handle both political issues and to lead a Middle Eastern Military Council. That left General Auchinleck to be able to concentrate on the military issues. He does seem to have focused to much on the dangers from the north from possible German movement into the Middle East from Russia. This is based on the account in Vol.II of the Australian Official History.