Inputting my 2 cents, and starting the argument all over, because I'm a masochist.
Assuming, as the OP does, that this happens after the surrender of Japan, we are left with an interesting scenario. Immediately after VE, the Soviets had a massive numerical advantage in Europe, though not a qualitative one. The result would have likely been a Russo-Japanese alliance, and years of war, with an eventual Allied victory. Waiting for Japan's elimination means that a sizable amount of Russian men and materiel had been transferred to Manchuria, but I still think they possessed a numerical advantage, albeit not as much of one. It also means no Japanese alliance.
Let's assume Russia is given 30 days to withdraw its troops to its pre-war borders, with the ultimatum running out on, oh, October 1. That would give the Soviets a month to prepare, and likewise with the West. This is also assuming that the West somehow finds the political will to do this, the most unlikely thing of all. Maybe Truman can use the Polish government-in-exile to officially request assistance, or Tito's actions in Albania and Greece as a pretext. We're still left with an intriguing situation.
America was definitely at a numerical advantage, but this advantage could easily be overcome, albeit in a very risky and potentially administration-toppling way: German troops. Many Germans harboured the dream, during the war, that if Hitler died Germany could join the West in fighting the Russians. Even Rommel believed it. German troops would be lining up to volunteer for service in kicking the Russians out of Germany.
With the manpower situation taken care of, equipment would be required. Seizing German equipment would help, but logistics is a serious concern in that 30-day period. That's simply not enough time to get all the necessary equipment from the US to Europe. Despite qualitative advantages, the West would be at a loss for materiel in the outbreak of hostilities.
Compare this to the USSR. Even with many of its troops in Manchuria, Russia still possessed more troops on the ground. Also, they were more fanatical than any of the enemies they'd be facing - except maybe the Germans - and accustomed to bloody, hard-fought victories. Psychologically, the Russians have the advantage. Qualitatively, many are poorly trained compared to the West, but they are often more experienced.
Despite this excess of manpower, the Soviets also suffered from acute logistical problems. They lacked equipment, and possessed poor transportation and communications. They sought to overcome this problem after VE by stripping German resources. this would enable them to overcome their materiel difficulties. However, those German resources had been stripped and sent back to Russia! They were no longer near the front line, where they would be able to overcome supply difficulties by shortening supply lines. Russia was faced with much the same problem as the West - a lack of materiel.
Many people have discussed air superiority. Here, the Russians actually possess a great advantage in Europe. They have more planes than the West, and Russian fighters are not that far behind the West in a technological sense. Unfotunately for the Russians, planes are one of the few things that the Americans can get to Europe within 30 days with shocking ease. They can simply fly the damn things there, stopping at Iceland en route to Britain. Both the US and Britain - potentially also France - can provide carriers to ease transportation as well.
Still, this is far more even than most people think, when it comes to fighters. In terms of bombers, America is in a distinctly advantageous position on both numbers and quality. However, most of America's best bombers would be in the Pacific - not somewhere they could get to Europe from quickly enough.
Politically, the Soviets have by far their greatest advantage. There are Communist Parties, most of them subservient to Moscow, and many of whom had long experience with guerilla activities against the Germans, in most countries of Western Europe. The biggest ones are in Italy and France. Corsica, due to the bumbling of French General Giraud, is under Communist control. These groups have great potential as fifth columns.
In addition to the Communist movements themselves, there are other left-wing groups. After VE, Staln order the small Communist Parties of France and Italy to amalgamate with other left-wing groups, particularly Socialist Parties, to form what were called "Popular Fronts." This means that, while Communist insurgents can disrupt transportation and communications in Western Europe prior to the outbreak of hostilities, the Popular Fronts can denounce Allied actions against the USSR publicly. These groups enjoyed a great deal of popular support. It is a well-known fact, for example, that the Popular Front may well have succeeded in gaining power in Italy if not for large-scale CIA funding of right-wing groups. As France and Italy were the two alternative fall-back points on the Continent for the Allies, this is a potentially dangerous situation.
In Europe, I foresee the war going on of two ways. Either Russia takes to the offensive, using its numerical superiority and fifth columns, combined with political unrest, to force the Allies out of Germany, Eastern France and Northern Italy. Eisenhower was still in command, and his contingency plan for a Soviet push called for Allied forces to fall back to Brittany, rather than Italy, meaning that any troops in Italy would be evacuated to Sicily and North Africa, then on to Brittany or Britain.
Turkey would acquiese to soviet demands in the Bosporus, and Greece would be invaded and conquered by Communist forces. There would be many guerillas active in both Greece and Turkey however, holding down Soviet troops that could be better used in the West.
After these initial Soviet successes, they would face even worse supply problems, while the Allies would dig in around Quimper - ironically, Charles De Gaulle's plan for France during the German invasion - and build their forces. They would then launch amphibious invasions in various parts of Europe, such as Calais and Italy, whittling down Soviet forces further. Eventually, when ready, full-scale invasions of Europe would begin, forcing the Russians further and further back, until they were thrown out of Eastern Europe.
Such an action would take many, many years, despite Soviet weakness. The reason for this, oddly enough, is that the West believed Russia to be omething on the order of three times its actual strength, which is a large part of the reason why America wasn't harsher with Russia after the war. Stalin exaggerated Soviet strength greatly, and the West, naively, took him at face value. In actual fact, the Russians had virtually no reserves in Siberia, and their industry and exconomy was devastated. an aggressive Russian move at the outbreak of hostilities would be designed, not offensively, but to give Mother Russia time to recuperate, and potentially imporve Russia's bargaining position in peace talks. It couldn't take a full-scale war, but it wouldn't have to fight one, as an aggressive Soviet push would have convinced Anglo-American analysts that their incorrect views of Soviet strength were correct, and they'd have bided their time and allowed Russia to slowly rebuild its industrial base.
The second possible outcome is a Soviet retreat. This is far more likely than the aggressive push, as the Soviets were actually much weaker than they implied post-WWII. Every time the Soviets were directly confront pre-1949, they backed down. In Iran, Turkey, and Finland, they pulled back. Stalin denounced Tito due to his aggressive support of Communists in Greece, in order to show Britain and America that he wasn't involved.
Stalin was afraid of the West in the aftermath of WWII. Contrary to popular belief, his actions in seizing Eastern Europe weren't aggressive, but defensive. He needed a buffer zone, and Eastern Europe was it. This was also behind his actions in neutralising Austria and refusing to allow German reunification, no matter the carrot offered. He was afraid.
The Soviets would not have retreated because of the ultimatum. Such an obvious display of weakness, in Russian views, would only have encouraged a Western attack - this view was possibly correct. Instead, as Winner stated earlier, they'd have begun a strategic withdrawal, taking everything they could of value with them as they retreated, and burning what they couldn't. They would have found a nice, defensible line somewhere, and planted themselves on it.
This makes their position harder to attack than in the first proposed outcome, as there's less of a front to attack, fewer partisans behind their lines, and a more limited Atlantic and Mediterranean coastline means aerial assaults from carriers and North Africa are less damaging. On the other hand, it also enables the Allies to bomb Russia itself. It potentially opens Moscow or Leningrad to nuclear assault, and, in the case of war, the US would have produced nukes far more quickly than in OTL. They might be unwilling to drop nukes on Russian-occupied France, or even Germany, as it could alienate their allies from those countries, but Russian territory is very different.
In the case of a retreat, I predict that the Russians will lose the war far quicker, but any sort of occupation of Russia is out of the question. Russians, recently having experienced Nazi atrocities, would take little convincing that the 'capitalist-imperialists' would do the same, and there's no way America had the manpower for the sort of pacification that would require. Even though a military victory was certainly possible, I don't see Russia's government falling, or the Baltics being liberated. I suspect that a treaty is signed which readjusts spheres of influence, with Russia achieving something comparable to its 1941 borders. No puppets states, but Tito isn't going anywhere, and would be more positively inclined to the Soviets in this situation. Albania is also likely Communist, and the 'aggressive' Western actions improves Communism's political position in Europe after the war.
As for in Asia, Russia had an absolutely mammoth advantage in terms of manpower. The IJA also still occupied many territories in Asia, even after the surrender, making it difficult for the US to mount any sort of offensive in China. The US did possess considerable naval and aerial superiority however, meaning that I don't see Russia advancing into Japan. I also don't see them wasting resources attacking Chiang Kai-shek in China. I instead see the Soviets ensuring IJA and and KMT neutrality, before stripping Manchuria of as much heavy industry as possible, much as really happened.
The interesting question is Korea. It would be difficult for the US to mount a successful invasion of the country, and many Koreans would welcome Soviet occupation if it displaced the Japanese. I see Russia taking all of Korea, and, due to the pre-eminence of the European sphere, the US not really doing much to dislodge them. America would be content to use strategic bombing to prevent any sort of attack on Japan.
Meanwhile, the Chinese Civil War would rage on, with an eventual Communist victory. After this, and with Vietnam freeing itself from the French quicker than in OTL, I foresee America getting far more concerned, and devoting far more efforts to dislodging Communism in Asia. Depending on whether the Soviets have been aggressive and met with cautious Allied action in Europe, or retreated and met with aggressive Allied action, America will either send a tonne of its own troops, or, much as it did with Germany, actually utilise Japanese troops. The latter is likely to cause them massive problems among native populations, and maybe even lose them Chiang, who had no love for the Japanese. Either way, I see Communism dominating mainland Asia, even moreso than in OTL. But Europe, Japan, and Formosa, will be capitalist.
My own prediction is for a strategic Russian withdrawal, as they simply didn't have the troops for an aggressive push, though there is alway the potential for them to do it as an incredibly risky bluff. As a bluff, I think it would actually have worked, due to Western misappraisals of Soviet strength. But I don't see Stalin actually risking it. I see Russia controlling the Baltic states and with a border further West than the one America demanded, but further East than OTL. All of Korea would be Communist, but, due to Russian weakness, I foresee it pursuing a far more independent course, and becoming more like Tito's Yugoslavia than Kim Il-Sung's juche utopia. China is much the same, though it likely splits with Russia even sooner.
The Cold War goes on, but with a far weaker Russia. Still, Communism as an ideological viewpoint would actually be stronger in the West, due to Western 'aggression.' I foresee it enjoying more political success in France and Italy than it actually did, and the establishment of pro-American dictatorships rather than democracies in much of Eastern Europe. Turkey would be less staunchly pro-American, due to the fact that Russia would not ba as big a threat. The formation of NATO is unlikely.
Everyone knows the Swedes are always impartial.
), numbers like 40 million are entirely possible. I don't say it is like that, but we know that the Soviets often intentionally misinformed the world about the casualties they suffered. Most likely, the official number of Soviet dead is underestimated.
