Ok but let me play devil's advocate here a bit.
First, there is some oil drilling going on near Cairo, right? So that is something right there. If there is already a refinery in Haifa that can be repaired too.
Well everything depends on
1. How well these facilities can be destroyed.
2. How long it takes to get to Haifa and Tripoli (Lebanon)
3. Just how much of the Med fleet is withdrawn through the Suez Canal and how much moved to Haifa (ships were withdrawn to there in 1942 when Alexandria was threatened by the German advance.
4. Where is the equipment to rebuild the oilfields and refineries coming from? So long as the RN retains subs and destroyers at Haifa, they can threaten supply routes. And don't forget Cyprus as an air base, at least short term.
Second, though those distances are indeed very long, and the Axis were limited in the amount of motor transport they had etc., they did manage to accomplish some extraordinary things in terms of projecting their armed forces quite long distances. Kursk is 1894 kilometers / 1180 miles from Dresden or Berlin, by road. Similar distance to Leningrad.
Budapest to Sevastopol is 2000 km / 1242 miles (again, by road).
Graz to Athens is 1670 km / 1037 miles
But they obviously managed to get a lot of men, weapons of war, materiel and supplies out to those distances.
Third, if there were already pipelines from Iran and Iraq, those could probably be repaired rather than built anew.
The main pipeline was as from Kirkuk in Iraq to Haifa with a spur to Tripoli (Lebanon) which was turned off after the French Armistice. There was no pipeline linking that to oilfields and refineries in the Gulf itself in WW2. There were also a dozen or so pumping stations along the route. Those would be less easy to replace. And 500 miles of pipeline are impossible to protect 24/7. That leaves it vulnerable to attacks by Special Forces (LRDG/SAS/SBS/Popski's Private Army were all active in the Middle East).
Fourth Abadan is 1000 + to Cairo over land. But if the Germans and Italians conquered Egypt, possibly the German or Italian navy could operate in the Persian Gulf and bring petrochemicals and oil through the Red Sea. Admittedly, that's a long and potentially perilous voyage, but might be easier to manage.
Do you really think that the Suez Canal will be captured without Britain making efforts to block it and mine it and to continue minelaying efforts thereafter? And to wreck the ports on the Red Sea Coast. In Spring 1941 the Luftwaffe mined the Canal and ships were sunk. It delayed the passage of Illustrious for about a week on her way to the USA for repairs.
There were a fair number of RAF bases spread throughout Palestine, Iraq, East Africa, Aden and into the Persian Gulf area, for retreating RAF units to operate from.
During the 1956 Suez Crisis the Egyptians successfully closed it off by doing precisely that. It was closed for 5 months and took an international effort , including by RN salvage ships, divers etc to reopen it. How quickly can the Axis assemble the resources to achieve that?
But you need to capture a lot more than just Egypt to make naval operations in the Red Sea / Arabian Gulf Indian Ocean / Persian Gulf area viable. You need to take Sudan, Ethiopia and the various British, French & Italian colonies in the Horn of Africa to secure the Bab El-Mandeb Strait giving access from the Red Sea to the Arabian Gulf.
The main Italian Red Sea base pre-war was at Massawa in Eritrea. It fell to the British East African Campaign on 8 April 1941, just 3 months after Rommel turns up at Tripoli (Libya). The East African campaign rumbled on until Nov 1941 when the Italians in Ethiopia and other Italian colonies were swept up. Those British forces will need to be pushed south to open the bottom of the Red Sea.
en.m.wikipedia.org
en.m.wikipedia.org
So long as a reinforced Eastern Fleet exists (with at least some assets coming from the Med) then the RN will seek to close off the bottom of the Red Sea to prevent Axis warships breaking out into the Arabian Gulf. The RN Eastern Fleet withdrew from Ceylon to Kilindini / Mombasa (Kenya) in April 1942. After the Japanese went to war the main job of the Eastern Fleet was to protect the western part of the IO. Axis movements like you suggest, represents the kind of threat it was there to counter.
And there you also have to secure both sides of the Persian Gulf to secure the choke point at the Straits of Hormuz.
You need to a map out to study. Again it demonstrates how WW2 was a true world war. The various theatres interact with one another. What happens in one does affect what happens elsewhere.
