Even in case of success the course of the war will be very different, depending if there is a VIchy-like goverment or a goverment-in-exile/Free Italy analogue (probably Balbo in Libya) the North Africa campaign is butterflyed away and the Meditterean is a British lake as the German Navy don't have the power to wrestle the control of this theatre from the RN, the Regia Marina will probably sail for Libya when/if Italy is occupied; the east-african theatre will not exist as the possibility that the Duke of Aosta follow the order of a quisling goverment are very very low.
I very much agree with Luke Dalton here. Yes, Italy is only....err....Italy. But even the Germans cannot do everything at the same time.
Contrary to 1943, the Italian Army wouldn't already have been beaten, and contrary to 1943, it would at least not be positioned on the wrong side of the country. Well, at least some of it.
So, Hitler has to decide which ally to give priority. Given that Italy is perceived to be the easier target and his rage about the Axis-partner's betrayal ("twice in one generrrration!"), I am very sure that there is a high probability for an "Italy-first" strategy for the spring/summer of 1940. Some general staffers might voice concerns that trying to go all the way down the peninsula might be a matter of overstretch, leaving open a multitude of flanks, but they will get brushed aside.
However, we should also bear in mind that before the battle of France, this is not yet the time for Barbarossa-ish Mega-Blitzkrieg-planning (a work in progress as a concept still), so the initial German campaign would be a plan to destroy as much of the Italian Army North of the Apennines and get hold of Italy's industrial core there before seeing what might be the next step.
France and Britain would try to move in à la Norway, ultimately not too successful probably, but the outcome would be better than OTL. France would at least bring itself into the very best frontline available on the Alps, but probably try to support the Italians in denying Torino and Genua to the Germans (though Milano is probably out of reach quickly).
With France and Britain still in the war, an Italian government would not be prone to capitulate too quickly (I can imagine it falling apart rather). In any case, the British have OTL proven that they would do everything to not let the Italian Navy get into the wrong hands. And as helpless as we would probably find them in stopping the Germans anyway on the mainland, as apt would they be to persuade the Italian colonies to remain on the Allied side. Same would go for Sardinia and Sicily. The latter might come into the German range of power-projection late in 1940, but battling that one out would lead to losses which will be felt later, especially when it comes to paratrooper-equipment. Sardinia would be a folly to attack for the Germans until they would have dealt with France successfully. And trust me, this timeline will not see an Afrikakorps.
Even if Germany subdues Italy within a few weeks during the early summer of 1940, the attack in the West has to be postponed. In order to be successful in Italy, Germany has to bring up -and risk- the best of its material. After that campaign, it has to be brought back into shape, reshuffled and so on. The campaign in the West would not occur prior to late summer or even later. This might be valuable time especially for France. Observing the campaign in Italy will push the French one or two notches up their learning curve in order to understand the German doctrines, especially if there have been encounters with German forces in Northwestern Italy.
The Sichelschnitt - operation was close to a gamble which paid off in an optimal way in OTL. It can only get worse for the Germans. So even if they prove superior to the Allies ITTL as well, they might be unable to break France before autumn and winter turn the campaign into a far bloodier and costlier campaign, encouraging the French government not to surrender, especially after the Med has been turned into a hard to break Allied stronghold over the course of the year.
And, please bear in mind, that is only IF France falls.
It will be a very different WW2.