I understand that the Allied forces originated in British-controlled Egypt, but how did the UK actually communicate with these forces?
What was the situation in the literal Mediterranean sea in terms of balance-of-power? I have read that the Royal Navy had dominance in the sea at the time, but was it to such an extent that movement of forces and communications was relatively easy?
I'm currently trying to wrap my head around the logistics of the situation around that time of the war.
Communication was easy. The British had pioneered the laying of undersea cables. And in the 1940s the empire was connected by a web of them. Telegraph and telephone wires ran from Cornwall to Gibraltar to Malta to Alexandria. So Wavell/Auchenleck/Montgomery could easily wire London and London could easily wire right back.
There is a good book Churchill and His Generals by Raymond Callahan that covers how all too often these generals felt they were a bit too accessible to Churchill and weren't able to manage their theater without his exercising too much personal influence.
Logistics was a bit trickier. Prior to the fall of France, Britain functionally had full control over the seas. But after that the Mediterranean became a contested zone. With the exception of operations to re-supply Malta (operation pedestal probably being the most famous) the British stayed out of the western med, at least until the end of 1942. The Eastern Med was safer, but with Axis control of Greece nowhere near safe.
So most supplies came to Egypt via Suez. For materiel that was coming from the commonwealth this was preferred anyway. No reason for Australian beef to pass through Britain. So that wasn't a problem. But the majority of finished goods still did come from Britain. And for them that meant a longer trip around the Cape of Good Hope.
U-boats were a concern. But not a large as you might think. The focus in 40-41 was on the mid-Atlantic gap. As the Germans were attempting to prevent supplies from crossing the Atlantic and reaching Britain. Ships leaving Britain and sailing for the South Atlantic were largely unmolested.
The trip was longer than the Mediterranean one would have been. But it was familiar and safe. And the British had more than enough cargo capacity to absorb the extra shipping time.
In comparison the Germans and Italians had a heck of a time supplying their forces in North Africa. They had a much shorter crossing. But the Italian merchant fleet was much less built up. And with Malta directly in the path of the crossing, and British bases in Gibraltar and Alexandria British planes and submarines could make the crossing very hazardous. And the Libyan ports weren't able to handle nearly the volume of good that the Egyptian ports could.