I remember reading a interesting article from the AMS a while ago about the Japanese mathematician Mikio Sato, who independently did some important work in algebraic analysis during the World War II. If my memory serves me well he was developing his theory of hyperfunctions at a young age all the while having to feed and protect his family during the war and "carrying coal" to earn a living. Here is a link to the AMS article: http://www.ams.org/notices/200702/fea-sato-2.pdf

Edit: Since it hasn't yet been mentioned, Alan Turing did great work during WW-II: he participated in a team that cracked the Enigma machine and many other codes/cyphers. https://en.wikipedia.org/wiki/Alan_Turing