I assume that most people may put focus on Polish or Soviet Union mathematical society when talking about this topic, but I personally would like to bring to attention another country which also severely suffered from WWII, namely China, through the experience of a talented, well-known, but low-profile mathematician, Wei-Liang Chow. The citations below are all taken from:
Wilson, W. Stephen; Chern, S. S.; Abhyankar, Shreeram S.; Lang, Serge; Igusa, Jun-ichi (October 1996). "Wei-Liang Chow". Notices of the American Mathematical Society. 43 (10): p.1117–1124.
Chow had ceased his research for about ten years due to WWII, according to S. S. Chern:
...The decline of Göttingen had the result of elevating Hamburg to a leading mathematical center in Germany. Her leading attraction was Emil Artin, the young professor who gave excellent lectures and whose interest extended over all areas of mathematics. Although WeiLiang was a Leipzig student, the German university
system allowed him to live in Hamburg. Besides the contacts with Artin, he had a more important objective, which was to win the love of a young lady, Margot Victor. They were married in 1936, and I was fortunate to be present at the wedding.
After their marriage Wei-Liang returned to China and became a professor of mathematics at the Central University in Nanking, then the Chinese capital. The next years China was at war, with the coastal provinces occupied by the Japanese. We next saw each other in 1946 in Shanghai after the war ended. In a decade of war years WeiLiang had practically stopped his mathematical activities, and the question was whether it was advisable or even possible for him to come back to mathematics.
According to Jun-ichi Igusa, Chow was able to communicate with European mathematicians during the first few years of his stay in China as a Professor at the Central University in Nanking, but then the situation became worse:
In the later years of our meetings, Professorand Mrs. Chow often mentioned the time when they were in China. After their marriage in Hamburg in July of 1936, they left Nazi Germany for China, and Chow started teaching at the Central University in Nanjing in September of that year. However, only one year later they found that China was no better than Germany. Imperial Japan enlarged a small fight on July 7, 1937, at the Marco Polo Bridge near Beijing to a systematic invasion of China. On August 13 a skirmish occurred in Shanghai, and on December 13 the “Rape of Nanjing” started. Fortunately they escaped Nanjing in September of that year to Chow’s birthplace, Shanghai. Shanghai being an international city, they felt safer there. They told us, however, that Shanghai at that time was quite similar to the Shanghai described in S. Spielberg’s movie, Empire of the Sun. In the first two to three years in China, Chow was still able to communicate with mathematicians in Europe, especially with van der Waerden. However, during the remaining eight years before he came to the United States the situation became so bad that he was unable to continue his mathematics. He told us more than once that it was Professor Chern who encouraged and helped him to come back to mathematics. Chow came to the Institute for Advanced Study in Princeton in March of 1947 and to Hopkins in the fall of 1948. He went on to say that without Chern’s friendship that might not have taken place.
But miraculously, Chow managed to return to his work after the war and:
His return to mathematics was most successful; I would consider it a miracle. He began by spending the years 1947–49 at the Institute for Advanced Study, after which he accepted a position at Johns Hopkins University, from which
he retired in 1977. At Johns Hopkins he served as chairman for more than ten years. He was also responsible for the American Journal of Mathematics, a Hopkins publication and the oldest American mathematical journal.
And many of his most prominent results, like Chow's moving lemma(1956) and Chow-Kodaira Theorem, were discovered after his return.