Handwavery ahead. Say the Axis is able to take Egypt in 1942 and is able to reach the Suez, would they be able to launch a Middle East campaign? Would the Arab population rise up in Palestine and/or Iraq? Would Axis paratroopers already in North Africa be able to use bases in Egypt to drop on the other side of the Suez in Palestine and achieve something or would they get slaughtered? At this point there is no Jewish brigade, that didn't happen until 1944, but the Palmach, the military arm of Haganah was officially raised by the British in mid-1941, roughly 1 year before Egypt was threatened in 1942, so they would be a factor, depending on whether the Arabs rise up against the British, then they might be forced just to defend their communities.
What would the US do? Would they still pull Torch or would they have to reinforce the Middle East directly and preclude landings in North Africa?