Norway: The armed forces stay in Norway for the time being, the Germans still have a foothold and could try something.A few thoughts :
Norway :
- The Norwegian Navy will be concentrated in the North Sea to protect the country. If they get enough forces, they might participate in the protection of the convoys from the UK to support the RN. They might also use their submarine force to block any raider (u-boot mostly) from breaking in the North Sea. Since the Kriegsmarine surface fleet is basically gone and their submarine force will be limited, the reconstituted Norwegian Navy might be rapidly up to the task with limited support from the RN.
- We already spoke about the air and land defenses of Norway. Basically, the Norwegian will be rapidly capable of taking care of themselves, with the support of a few allied units. On the other hand, the use of Norway as an offensive base will take time and will require RAF bombers and an improved infrastructure.
- The Norwegian merchant marine will probably still be the biggest contribution of Norway, as OTL. ITTL, we have a major difference with continued access to Norwegian manpower to man the ships. The limited access of the Kriegsmarine in the North Atlantic and the North Sea might lead to less merchant losses. If Barbarossa comes around, the convoys to Northern Russia will run essentially unchallenged.
- I don't think we will see any Norwegian military unit outside of the country or the waters around it. The exception might come with training for specialized units or (pilots, new ships bought abroad, tank, marines and parachute forces, ...).
Sweden and Finland :
- OTL, those two countries were essentially cut from the rest of the world by the German conquests in Norway and Denmark. ITTL with Norway free, they still can trade with other countries through land (Norway in Narvik but also Oslo and then Bergen) and possibly sea (there will be some neutral convoys run by Sweden in the North Sea).
- The UK will probably try to buy as much iron from Sweden as they can, forcing Germany to pay much more for their. This give a bonus in saving shipping for the Allies since Sweden (along with French North Africa and Spain) is far closer to the UK than alternative sources like the US and Canada.
- Continued trade might give enough political liberties to Finland so that they won't feel the need to participate in Barbarossa.
France :
- The French government have decided to continue the fight. They come with the whole Empire, the Fleet, the merchant navy and the gold, plus whatever they managed to pull out of mainland France. And will be France using the gold to buy US arms and planes.
- Depending on when they made the decision, whatever they managed to pull out of mainland France can mean a few hundred thousand mix of soldiers and civilians to almost one million people with some heavy material and specialized workers (from the southern parts of France). OTL, the majority of the Armée de l'Air was withdrawn to North Africa by the time of the Armistice. So, at first, the problem might not be the lack of planes, but a lack of spare parts.
- Outside of the Courbet class battleships, the French fleet was pretty decent and adapted to fight in the Med. It's main weakness was in anti-submarine warfare, but, with the help of the RN and air power, the Marine Nationale will be a big player against Italy. The Richelieu is about to enter service, if need be, the French will use parts from the unfinished Jean Bart (which in turn might be finished as an aircraft carrier).
- At least at first, the French will probably be able to keep Corsica, since the Italians don't have the airborne or the amphibious capacity to invade. As long as the Luftwaffe don't deploy en masse in Southern France and in Italy, the French can use the island as a way station for boats and planes which don't have the capacity to reach North Africa.
- Tunisia is a good base to support Malta in blockading Italian convoys to Tripoli (2/3 of Libya's port capacities).
- The French North African Army is far from being the best army on earth, but they will force the supply starved Italian Army in Libya to fight on 2 fronts. And, with Tripoli only 80 km from the Tunisian border, the port is in range of the French reinforced with with evacuated forces from France and what ever the US will spare. The most important US contribution here won't the tanks (the M2 were abysmally bad), but the trucks, the old equipment (used OTL to arm the British Home Guard) and ammunition. Once Tripoli is in Allied hands, the North African Campaign is won for the Allies.
- There was one French Corps in Syria which was intended to support Balkan countries if they choose to support the Allies. Those forces can be transported in Egypt to support the British offensive against Libya from there.
The UK :
- With Norway in Allied hands, the RAF will be able to concentrate in the South of England. Plus, with the Kriegsmarine surface fleet nonexistent and France still in the fight, the invasion scare of 1940 will be widely reduced. So the UK industrial mobilization will be more efficient.
- With the Kriegsmarine surface fleet nonexistent and Norway in Allied hands, the RN will concentrate on blockading the U-boots coming from the French coast and Germany. I don't think the Home Fleet will have that much heavy units during the war, but mostly escorts and cruisers forces.
- With the lesser losses in aircraft carriers for the RN, they will be able to make a 3 carriers attack on Tarente (possibly also earlier). This might mean that the Italian surface fleet won't be a problem for long too.
- With North Africa on Allied control early on, the Allies will probably be able to run fast convoys in the Med all through the war under the umbrella of Allied aircraft all the way. And Malta won't be such an isolated base.
- It means that the RN will have the possibility to concentrate its forces if Japan makes a move in the Far East.
Germany :
- Germany, despite its failure in Norway, is now in control of Western Europe. I think AH will still try to force a British surrender (or negotiated peace) with a Battle of Britain, but ITTL there is even less chances of success.
- AH still has a non-healthy obsession with the USSR so I'm pretty sure he will launch Barbarossa, preferably in 1941 as OTL. But, to be able to do so, the Nazis will need to secure their Southern flank. So they need to take Corsica in 1940 or early 1941. They probably also need to sent so Luftwaffe forces in Southern Italy to attack Allied convoys (and block any possibility of Allied landing in Sicily). And, depending on Italy's actions, the Nazis might need to secure Yugoslavia and Greece as OTL.
Italy :
- Italy is in deep problems. They will almost certainly loose Libya rapidly. East Africa is isolated and will fall at one point or an other. Their Fleet is desperately outnumbered by the Anglo-French (and it might be sunk at anchor pretty quickly). And, the bonus is that the Dedocanese Islands are isolated by Greece.
- The big question in 1940 is if Benny is dumb enough to invade Greece as OTL. If yes, the Allies will have the forces to reinforce Greece in early 1941. If not, he might loose the Dedocanese Islands at some point in 1941.
Japan :
- Japan is still deeply involved in China. But, contrary to OTL, they won't be able to invade Indochina without the UK getting involved and declaring war. So, war materials will continue to flaw into China from Haiphong.
- I still think that Japan is on a collision course with the US in the region, but the butterflies are flapping so it's hard to predict.
Essentially you are right, but it means war between Japan and the UK the moment Japanese forces invade. The British can't afford to let Indochina fall or let the French Colonial Empire fall apart for political and strategic reasons. So you might have an escalation a year earlier in the Far East.
I'm not sure I get it. Where is the two fronts war for Germany ? They just kicked the Allies from the continent in France and they have no hope to come back until 1943 at least (without the US).
Sweden+Finland: They won't be as trapped and pressured as OTL.
France: No French surrender memes (dang!), they can free up a lot of British assets, especially Naval, from the Med, and will be using their gold reserves liberally.
UK: Not as stretched, less defeatism, Japan won't have it as easy.
Germany: BoB will still happen, the next German plans concerning Norway will come soon.
Italy: Still in war as OTL, will have a harder time
Japan: This somewhat butterflies OTL, I'm not really going to cover it (this is a Norway TL after all)