@wiking
I don't understand why you consider that France's Fall is a given in a scenario with Italy in the war on the Allied side since 1939.
Per OTL with Italy in the war it is impossible, France won't surrender like IOTL with Italy to retreat to if the Germans breakthrough. That said it is unlikely that Italy alone would butterfly the May 1940 breakthrough.
And we do have to consider that Italy may make the Allies too overconfident and Operation Pike happens ITTL:
https://en.wikipedia.org/wiki/Operation_Pike
1. The Allies have far more units on Theater than OTL :
The French have the units from the Armée des Alpes (it's two Corps and five first line Infantry Divisions even if you disregard the four Forteresse Sector and the F-17 tank battalion) and from the Armée d'Afrique (Without the forces in Morocco and the garrison troops, it's two Corps, four or five second line Infantry Divisions, one Cavalry Division, a number of Cavalry regiments and two D1 tank battalions).
The UK have, at least, one more Infantry Division from the Middle-East and they might have the 1st Armoured operational by taking personnel and equipment from what will become the 7th Armoured.
I don't know what the Italian will sent for sure, but I think a five division expeditionary force is a minimum with one Armored, one Celere and three Infantry Divisions in two Corps.
That's a non trivial amount of forces (16+ Divisions) and most will be in strategic reserve.
You're literally inventing forces that didn't exist as combat units. All the stuff left in North Africa/Syria at this point wasn't combat ready in Europe and needed to be kept in North Africa for training and garrison duty regardless of Italy. Other than the Alpine forces nothing in Africa that wasn't already stripped out IOTL was fit for fighting in Europe. Besides with the Italians and the Alpine army, there wouldn't really be a need to bring even more less than worthwhile forces from the colonies home. Especially the shitty D1 tanks; they were exiled to the colonies for a reason.
In terms of the Alpine forces redeploying the 5 divisions/2 corps would be reasonable, even the FT-17s.
The question is where they would be deployed if not on the Italian border; I think considering they were building up forces in Syria to land in the Balkans they may well send those forces to Syria:
https://en.wikipedia.org/wiki/Polish_Independent_Carpathian_Rifle_Brigade
On 12 April 1940 the
brigade was officially formed in
Syria, with Colonel Kopański as its commander. The main base of the brigade was established in
Homs and the new unit instantly entered the ranks of the French
Armée du Levant. As a unit specializing in
mountain warfare, the brigade was thought of as a Polish addition to
Allied plans for landings in the
Balkans.
Mountain and colonial troops were needed for the planned Balkan operation. Would be highly interesting if the French got the necessary forces along with Italy to land in the Balkans in Spring 1940 before the invasion of France to put pressure on Turkey to join the Allies...
That and where and what the Italians would send. I think an armored division is a good bet, but perhaps something like the Italian expeditionary corps in Russia with an extra armored division:
https://en.wikipedia.org/wiki/Italian_Expeditionary_Corps_in_Russia
The forces the Brits had in the Mediterranean in 1940 before Italy entered the war were needed in the region anyway to hold it down and potentially deal with Russia.
In the air, the French will have more units in Northern France, kipping only reequipping units in the South. It's not much, but it's something.
2. The German forces are more stretched than OTL :
Even if, like you said, the German don't need much forces to cover Austria, they will still need to deploy more forces there. It will be probably five to ten Infantry Divisions, mostly third line with maybe one or two first line.
What significant extra air units will the French have?
And why do you think it would take more than 5 division to hold the Brenner Pass? 5 low quality ones that weren't doing much else anyway? Frankly Volkssturm with WW1 weapons could hold the pass.
Plus, the Luftwaffe will need to cover Austria. With 20/20 incite, we know that the Italian Air Force was a paper tiger, but the Luftwaffe don't know that for sure (They might suspect it). So that's some Fighters (maybe 1 or 200) staying in Austria.
That's some (minor) more forces tied up outside of the Western Front. It's not much, but those will be missed, specially if the French Front isn't pierced immediately.
3. You can't squeeze much more forces on the Dyle-Breda Maneuver :
OTL, the British were already making noise that they didn't have enough terrain to deploy correctly with the 7th French Army on their left.
4. The Allied reserves will be more effective :
OTL, the French reserves were split in four parts : 1 to support the forces in Belgium, 1 to support the Maginot Line, 1 to cover the Swiss option and 1 central in Champagne.
Given the quality of the Italian air force and it's need to commit forces to France, potentially the Balkans, and defend it's own territory, it is entirely possible to defend Austria with little more than FLAK, as at the time bombing of cities wasn't the policy of the Allies. The French refused to even bomb Germany for fear of retaliation, the British did the same after the heavy loses in their 1939 Kiel raid, as well as hope to avoid bombing of cities in general:
https://en.wikipedia.org/wiki/Phoney_War
https://allthatsinteresting.com/phoney-war
Not seeing how and where the extra Allied forces would be deployed in the West other than as reserves, which per OTL were not effectively used anyway. Not only that, but the extra forces may well end up used in a hare-brained operation in the Balkans as the French were quite serious about it and had an expeditionary corps building up in the Levant for such an operation; here they don't even need to form those forces, just use their Alpine Army for the operation, while having the extra air units with Italy in the war to bomb Soviet oil as they had planned to do IOTL, but now have every reason to not wait on thanks to Italy removing the need to maintain a reserves and actually adding forces for a potential operation.
OTL, the Luftwaffe effectiveness against the reserves was magnified by two factors.
First, the French reserves came piece-meal. It's a strategic and doctrinal mistake from the French High Command which might only be change on the margin : with more reserves directly at hand, Gamelin and Georges might be more willing to engage more rapidly and massively the reserves.
Second, after the first few days of fighting, the reserves came form afar (mostly from the Maginot and the Swiss pools). They were dropped disorganized at some train station, needed to concentrate and reorganize, and then walk into the front threw predictable path. All that under the total air dominance of the Luftwaffe.
I suppose that the "new" ITTL reserves will mostly be deployed in Champagne, including the Italian Expeditionary Force. It's central and it have the military infrastructure to accommodate those forces (OTL, the 7th Army was stationed there before the Breda variant was chosen).
5. It can snow ball :
I'm aware of the advantages of the 1940 German Army OTL, but even with them OTL, the Battle of France was far from a walk in the park with some very close calls (the Landser who took an unfinished bunker in Sedan making it possible to roll the line, the two survival of Rommel, the multiple times were German generals disregarded the orders to stop during the Sickle Cut, ...).
If you take into account all those changes, the German breakthrough is not a given at all, simply because the Allies have organized reserves in place.
Likely the extra reserves get used for the planned operation in the Balkans with Italian assistance and operations against the USSR's oil.
The French weren't exactly keen on fighting the Germans head on in Europe, hence the Phony War, planned Balkan operation, and planning bombing of Soviet oil. They wanted a peripheral strategy and actually having the reserves to launch those before the Germans could invade France may well mean they do to distract Germany from invading France, but in the process end up screwing themselves badly.
If we are going to talk about butterflies we need to talk about ALL of them, not simply the ones you think will give the French the best change with minimal changes to OTL, as well as the understanding that per OTL the French may well still shit the bed when it comes to using the extra forces they have even in France.
On the Norway Campaign, the major change I can see is that the Allies have even more naval assets. After all, the Mediterranean Fleet and the French Fleet were deployed to deter Italy OTL.
So I can see ITTL an even bigger Home Fleet, possibly reinforced by the French Force de Raid, deployed in three squadrons of two battleships and one aircraft carrier and the two battlecruiser squadrons.
By adding some ships, you dramatically increase the chances of interception of the German ships, specially for the Trondheim and Narvik invasion groups.
The Mediterranean fleets were not meant to operate in the Atlantic or North Sea, where conditions are quite different. The Germans even found out the hard way that their North Sea designs for capital ships were not sufficient for Atlantic conditions, which required major modifications for an 'atlantic bow'.
Likely the Italian fleet and French Mediterranean Fleet get sucked into supporting a Balkan operation, as the French had planned and was still building up forces for in June 1940 when they were defeated IOTL; ITTL they will have the forces thanks to Italy being on side, so may well launch it in 1939 or early 1940.
Besides the Norway Operation was a Churchill scheme which the French would only contribute a limited force for, there is no reason to involve more French forces, as the British fleet was enormous and running nearly the entire thing. The French had their own plans and could leave the British to run theirs. Your reasoning for the enhanced Narvik operation is entirely based on hindsight and not the actually Allied views and plans at the time, which Italy wouldn't appreciably change, other than perhaps France being able to launch their peripheral strategy in the Mediterranean from the get-go. In fact I wonder if they might try to involve Romania in the war with their Balkan strategy.