At that point it was pretty certain that Norway and Sweden would offer free passage for the troops. The background was that the League of Nations had unanimously declared that the war was an act of aggression by the Soviet Union and both Sweden and Norway, as members of the league, felt it their duty to help in a coordinated league effort, which would be de facto the case if France and Britain united and asked the other league members for support.
So for Sweden and Norway, the motive would be solidarity. For England the motive was apparently that a joint effort with S and N would bring them closer to them and less close to Germany. France's main reason was that if there is a war on the continent, it would better for them if it happened as far from home as possible.
Solidarity is all well and good, and Sweden sent a lot of material aid and volunteers with weapons and even aircraft to Finland during the war. Also the Norwegians sent help and volunteers, as did the Danes. But what we have to remember is that especially Sweden also did everything possible to keep this help unofficial and to stay out of the war officially.
I bolded part of your post above because it directly contradicts OTL facts. When the potential Anglo-French intervention was discussed in early 1940, the Swedish government consistently insisted that it will not allow the passage of foreign troops. To Finnish government representatives, who visited Stockholm several times during the war, the answer was always the same: Sweden will not allow foreign troops en route to Finland on its soil. When the situation culminated in February - March, I believe the Swedish went as far as to say that they are ready to use military force against such foreign troops that enter Swedish soil without the government's permission - a natural stand for a neutral nation to take, and one that made the situation clear to the Finns.
So without a pre-war POD or several, I am sceptical about the OP about the formal request for support happening. The Swedish desire to stay out of the war, a tried and true policy that had seen the nation through WWI and had a 130-year history behind it at that point would be hard to overcome. The Swedish government's policy was one of the two major reasons the request for help was not made - Finland did not want to take the responsibility of plunging its neutral neighbour in a world war against its will. The other reason, of course, was that any realistic help would have been too little and too late at that point.
As to answering the question in the OP itself: if Finland makes the request and for some reason both the Norwegians and the Swedish do allow passage to the troops, Stalin would have all the incentives needed to make an early peace with Finland. This might lead into him losing face (even more than IOTL), but it would have to be necessary to avoid one of his greatest nightmares: a potential war against a unified front of capitalist powers if now Hitler would have made peace with the Allies and turned on the USSR instead (like Stalin feared).
Even the launching of Operation Pike might not lead to Stalin declaring all-out war on the Allies. Sure, the USSR would milk it for all it is worth in its propaganda. The capitalist powers turning on the Soviet state and people in support of a White Finnish bourgeois-military junta. A bloody, unprovoked attack by the British on the peace-loving proletariat of the USSR. The German propagandists would also have a field day, playing up the belligerence of the Allies and the danger they present to all neutral nations.
The hot war in Finland would be wrapped up, with Finland still independent and possibly with even more land than after the OTL Moscow Peace. This might well happen before the Allied troops really make their presence felt on the front. This would have been in the interest of both the Soviets and the Finns - we know the Finnish Army had no reserves left by March 1940 and was only weeks away from a potential collapse of the front. If Finland sees that only a trickle of Allied troops would be reaching Finland (more staying in Sweden to "secure" the iron ore mines), the government and military leaders would have really no chance than to accept a peace offered by Moscow. Now, helping Finland was the casus belli - after Finland and the USSR are in peace, and even if it is an uneasy one neither side wanted, what would be the reason to keep an active war against the USSR going when the Germans are a much more imminent danger to France and Britain? And where and with what troops would that war have been fought?
This is to say I believe that in 1940-41 we might see a Allied-Soviet "phony war" instead of a hot one, especially after the fighting picks up in France (and after a potential Fall of France) ITTL.
The plus sides for the Allies would be that they would control Norway and that even Sweden would now be seen as a de facto Ally. The potential (but IMO not really inevitable) further German-Soviet rapprochement in 40-14 would be a risk the Allied leaders would have to think very hard about, but I think the natural mutual distrust between Hitler and Stalin (as well as the ideologically opposing nature of the Nazi and Soviet systems) and the highly likely German attack against the USSR by 41-42 would suggest to me that this would not be a TL where a Nazi-Stalinist behemoth takes of over Eurasia to fight hand in hand in an extended war against Britain and the US like some seem to think.
