On multiple occasions in WW1, the British built RR at the rate of 1 mile per day. So 500 km per year is technically feasible. So looks like you could largely fix the Libyan logistic issues with a POD in late 1939. But I guess we would need Benny to be a better administrative and military mind.
I suppose the very limited OTL rail met the commercial needs of the colony, and the Italians were not thinking they would get in the kind of war that they would need a high capacity railway. So the earliest POD would be September 1940 when the Italians figure out that the British are not going to make peace.
By then you are talking a lot of steel to cover long distances, steel useful for other things for countries at war.
I wonder since there is a coastal decent road if something like the modern Australian road trains could be improvised here to deliver lots of supply without rail: Basically the engine of a diesel locomotive, but with road wheels pulling multiple trailers on a paved road.