Everyone is talking about the French sweeping from Tunisia into Libya like it would be a walk in the park, but it wouldn't. The Italians had concentrated most of their assets in Libya on that location AND they had a short rail link to their main supply hub in Tripoli conveniently located near the border.
Meanwhile, the French had the fortifications of the Mareth Line they had built in case of war with Italy located some km further west :
Link
The line had been built by the French in pre-war years to meet possible Italian threats from
Tripolitania. In the coastal plain, here from ten to 15 miles wide, it ran immediately behind the line of
Wadi Zigzaou from
Zarat to Touati, the wadi being a natural anti-tank obstacle with sheer banks reaching in places a height of seventy feet. This stretch was moreover covered along the whole length by concrete and steel pillboxes and emplacements which were in existence before the war.
From Touati the line swung south-west to a point just south of
Toujane, and then north-west through the
Matmata Hills towards
Djebel Melab. In the hills the nature of the country was relied on for defence, and there were few artificial aids.
In pre-war days it was considered that the
Mareth Line could not be outflanked, because the Dahar was thought to be impassable for mechanical transport. In 1938 a French lorried force carried out an exercise to determine whether such an operation was possible, and came to the conclusion that it was not.
So, we have the French badly beaten up in their own homeland and severely disorganised. Most structures will be ad-hoc improvisations, something the French military was not really good at at the time. Moreover, they are in their prefered position: A fortified line they think cannot be flanked.
My guess is the French stay put, and the Italians advance the hundred and something km that separate the line from the border, and build their own defences opposite it after a failed half-hearted attack or two. Neither side is the able to break the stalemate until the British sweep in from the east, since, if the French attack, they would be facing Italian troops doing the only thing they know to do reasonably well: defending fortified positions in hilly terrain with no fear of flanking attacks.
Regarding the British offensive, I think it will be a bit slower than OTL.
On the plus side, a lot of the Italian forces are busy in the west building trenches opposite the French.
On the downside, Italy could probably spare more troops for Lybia, since getting stuff from the ports to the front was the main issue, much less than port capacity. With their units in Tunisia a stone throw away from Tripoli and with half that distance covered by rail, the Italians should have the needed trucks to supply new forces to be deployed east against the British. Moreover, these troops would not be caught out of position at the end of their supply line deep into Egypt, but rather behind pre-prepared coastal fortifications.
Thus, the British would probably face a series of somewhat more bloody and a bit more lenghty sieges, probably dragging on the conquest of Cyrenaica into March, or even April or May 1941 instead of February as OTL, especially if Balbo survives.
Now, the thing is on February 9th OTL, Churchill ordered a halt to the African offensive and shortly after had troops shipped to Greece. Question is: If his forces are still east of Benghazi in February, and the French are still in the war, would he still go ahead with the operation in the Balkans to impress the Americans ? My guess is yes.
If this happens, then British offensives in Lybia are halted and a lot of the troops are sent to Greece, where they are defeated by the Germans.
If Churchill refrains from Greece, he could probably take Lybia in short order as the precarious Italian position might collapse. Although, as said, I find this unlikely.
This means the British would be ready for another go at the Italians only after recovering from their Greek debacle in mid May, when they launched OTL Operation Brevity, or even as late as mid June, when they launched Battleaxe. Thus, the fall of Lybia should become apparent to Mussolini only after Barbarossa has been launched.
Would Hitler form the Afrika Korps with much needed tanks and trucks now that the ultimate struggle of the Arian race has begun ? My guess is no, which means the Italians loose Africa by early Autumn 1941 at the latest.