Despite desperate pleas from the Swedish government, the Finnish Parliament approves the request of aid on March 1940, hoping that the promised 50 000 troops French and British governments claimed to deliver by a mandate from the League of Nations on March 1940 on few weeks could somehow force Stalin to offer better terms. Instead, the Soviet dictator urges his generals forward to Helsinki.
While Allied bombers from Syria try to raid Baku oil fields and a token presence of Allied troops actually arrives to Lappland and enters to combat against the Red Army, most of them firmly entrench themselves to northern Norway and to the iron mining areas of northern Sweden. Hitler, dismayed, invades Denmark, Norway - and Sweden, officially to liberate the latter two from Allied occupation. As the Soviet forces crush the exhausted remrants of the Finnish Army, the country falls to guerrilla war and the few surviving Finnish units withdraw towards the Swedish border along with the surviving British and French forces.
And there you have it, Finland in the Allies.