Here are some Soyuzmultfilm films from the 1940s that I subtitled. The first is "Elephant and Pug" (or "Elephant and Moska the Dog"), which was made in 1941. Directed by Lamis Bredis and Pantaleymon Sazonov, and based on Ivan Krylov's children's poem of the same name. It's about a little dog who tries to seem tough by fighting in a big fight which is completely safe for him.
If subtitles appear to be missing: Click on the triangle at the bottom right of the video, and make sure that CC is turned on.
Next, "The Stolen Sun", made in 1943 (if you believe animator.ru) or 1944 (if you believe the date in the film). Directed by Ivan Ivanov-Vano and Olga Khodatayeva. It was also based on a previously-written children's poem, this one by Chukovsky. The idea of the film is that the best things in life must be fought for.
"The Stolen Sun" was made in the middle of a horrible war, which was (speaking strictly empirically) around 100 times more deadly for the Soviet Union than for the United States. Many animators who had been working at Soyuzmultfilm were sent off to the front lines. The remainder (who were generally the better ones) were evacuated to the Ural Mountains, where they made just a few short animated films per year. However, even from 1939 to 1942, production was much less than in prior years. I guess the country was getting ready for something. Look at this:
1937 (the year of Soyuzmultfilm's founding): 24 films
1938: 28 films
1939: 10 films
1940: 8 films
1941: 10 films
1942: 5 films
1943: 4 films
1944: 7 films
1945: 4 films
After that, the number of films made per year slowly increased, but only got to its pre-1939 level after Stalin's death.
Despite the small number of films that were made, however, they were made extremely well. "The Stolen Sun" is really glorious, with skilled direction, very detailed animation on ones, and a superb orchestral score. The excellent animation of birds and butterflies in both of these films is definitely something to be studied. The future director of the Russian Winnie-the-Pooh films, Fyodor Khitruk, left the studio during the war to work as a translator (he had lived in Germany in the 1930s), and came back after to find that the expected animation quality had become much higher, and that his skills were no longer sufficient (source: his recent book).
Soyuzmultfilm also made propaganda films during the war. These were called "film-posters". I'd say that they constituted about half of its production at the time. Here are a few (links go to videos with subtitles):
Fascist Boots Shall Not Trample Our Motherland (1941)
Cinema Circus (1942)
Journal of Political Satire No.2 (1942)
To compare: American animation studios also made obvious propaganda films at this time. Most notably, from 1941-1945, Disney was under government contract to produce 32 animated propaganda shorts (source), and Disney also funded some propaganda on his own initiative, such as the feature film Victory Through Air-Power.
Soviet animation never combined the genres of children's cartoon and propaganda cartoon, even during the war, which is something that American studios did do (some examples: Donald Duck living in Nazi Germany in "Der Fuhrer's Face", Popeye the Sailor in "You're a Sap, Mr. Jap", the Three Little Pigs in "Blitz Wolf"). The Soviet children's cartoons during that time very often had dark, timely themes, but there were no direct references to current events or crude anachronisms in fairy tales; these films can viewed independently of their historical context. On the other hand, children's cartoons in the Soviet Union were meant to serve a useful purpose by shaping children into good adults, not just to be fun. For more on this, see an earlier post in which I talked a little about it...