Thucydides Redeemed
It's kinda easy to have the Winter War end up as a more comprehensive Soviet victory as despite the heroism and tenacity of the Finns almost all of the reasons for the Soviet humiliation can be pinned on the errors of Stalin and the Red Army. ITTL the dreadful Soviet plan of invading across a broad front with only two weeks ammunition is put in the bin as it should have been IOTL, instead this world has a Soviet armoured spearhead with a singular line of advance down the Baltic Coast and plentiful ammunition and fuel hitting the Mannerheim Line as a speed bump. Despite the Finns best efforts there's not much you can do when your army only has four weeks ammunition and your enemy is fighting the sort of war it's prepared for. Swedish aid is inadequate and Anglo-French reassurances turn out to be diplomatic niceties and nothing more.
The Red Army breaks through the Mannerheim Line on Christmas Day and by New Year the Finnish bourgeois state has largely unravelled; the country is on the move and the remnants of the Finnish army have largely ceased resistance in favour of returning to their families or assisting refugees on the arduous trek west. With nothing left to hold off the Red Army and the President showing up in Stockholm it's left to Sissi Wein, the Finnish Vera Lynn, to defiantly sing Vapaussoturin Valloituslaulu over the Yle radio waves the same day the Red Army marches into Helsinki.
Finland emerges from the conflict, whether one calls it the Winter War, Second Finnish Civil War, or even The State Capitalists Imperialist Aggression Against The Finnish People, locked in the vice that is Moscow's loving embrace.
Prime Ministers of Finland
1937-1939: Aimo Cajander (National Progressive)
1939: Risto Ryti (National Progressive) [1]
Chairmen of the People's Government of the Finnish Democratic Republic
1939-1940: Otto Wille Kuusinen (Communist Party of Finland) [2]
Chairpersons of the Presidium of the Supreme Soviet of the Karelo-Finnish Soviet Socialist Republic
1940-1953: Otto Wille Kuusinen (Communist Party of the Soviet Union)
1953-1967: August Wesley (Communist Party of the Soviet Union) [3]
1967-1974: Hertta Kuusinen (Communist Party of the Soviet Union) [4]
1974-1987: Noora Latavia (Communist Party of the Soviet Union, then Communist Party of Finland) [5]
1987-????: Kimmo Rentola (Communist Party of Finland)
[1] The respected economist and politician was called to lead his fatherland through the greatest crisis in its short history yet when it became clear the odds where genuinely insurmountable he fled with his cabinet to join the President in Stockholm, waiting for an Anglo-French declaration of war on the Soviet Union to save his country. It never comes and worse still the Swedish annexation of the Aland Islands with tacit Soviet agreement causes the government-in-exile to leave Stockholm in protest. The move to Paris and then to London takes its toll on Ryti as he becomes convinced of what only a few people actually believe, that he was a coward who fled his country in the time of need on a deluded quest for foreign help. Resigning as Prime Minister in 1940 he avoids the sidelining the Government-in-exile faces after Barbarossa and their following descent into irrelevance as the Cold War begins, remaining prominent in the Finnish exile community until his death in 1956.
[2] Already a controversial figure before the Soviet invasion, Kuusinen has the distinction of being the father of the FDR and its executioner. Reassured the Finnish people that the Red Army had arrived to help assert true Finnish independence from the White traitors then signed off on Finland's incorporation into the Soviet Union a few months later. Introduced aggressive land reform and housing campaigns that worked out quite well economically but led to devastating deforestation, sanctioned the harsh NKVD counter-insurgency campaign against the IKL during the Great Patriotic War but managed to convince Stalin to ease off on mass conscription of Finns in favour of a Finnish Solidarity Front. Made up of old civil war leaders with a mix hand picked fanatical Stalinists and genuine volunteers, they play a key role in throwing the Germans back from Leningrad in September 1941. Benefits from being responsible for the one area of the western Soviet Union that wasn't destroyed, facilitating reconstruction whilst also the K-FSSR's own economic development. Doesn't survive De-Stalinisation.
[3] The Civil War and Great Patriotic War veteran is reluctant to take on the responsibility of leading the K-FSSR, citing his age but realistically because he never supported Finland's annexation into the Soviet Union in the first place. Tries to do his best with what power he has, embracing Khruschev's establishment of individual economic plans for each SSR. Introduces some voluntarism and gift economics into the Finnish economy as a sign of recognition that the country still isn't the urban paradise of Kuusinen's dreams. Repeatedly applies to become ambassador to Havana until he is old enough to retire even by Soviet leadership standards.
[4] Not as controversial as her father and suddenly much better connected in the wake of Khruschev's fall from grace, Hertta Kuusinen moves out of Otto's shadow fairly quickly to drag the K-FSSR out of its sedentary cosiness and into the modern world, this involves Soviet nuclear technology finally being introduced and East German childcare to abolish the joint evils of the cold and the patriarchy. Her personal role in assiting Suslov's coup against Andropov gives Finns a certain amount of national pride back, their country is punching above its weight and its increasingly metropolitan. Some haggling allows Finland to keep its own economic plan even after the initiative is largely wrapped up for much of the rest of the USSR and the knock on effects begin to be shown as internal migration towards Finland brings new engineers and scientists whilst exiles from Sweden begin to return in not-unremarkable numbers. Finland has a genuinely popular Chairperson at long last, at least with the Finnish people, but Kuusinen's role in broader Soviet politics has made her ignore the rise of the "Baader-Maoism" that has already begun to affect much of the Eastern Bloc. This is politically fatal as the Soviet Union becomes the last domino to fall.
[5] A prodigy of Kuusinen's who eventually ended up stabbing her in the front at a meeting of the K-FSSR Supreme Soviet, this orphan of the Winter War was raised in the Stalinist worldview which, taken to heart, made much of the socialist world as it was rather incongruous. The success of the RAF in West Germany becomes infectious in the east and it isn't long after that young people in Finland are using the confines of the CPSU to debate "actual" Marxism-Leninism for a change.