Sure, but World War II started for America in December of 1941, with the attack on Pearl Harbor. The vast majority of people here didn't give a rat's ass about the invasion of Poland or Hitler or anything else happening in Europe until Americans started getting killed. That's just the way it was, and I'm sure that it was not only Americans who were guilty of having blinders on when it came to horror and atrocity being visited on "someone else." So American "Troops" wouldn't have been pulling innocent people off buses and beating them until 1942. Assuming that even happened, which is doubtful.
Bukowski did not leave Los Angeles until 1942, I think we can say that with some certainty. Exactly when in 1942, we don't know. Though it was probably early, and it was probably at least partly in reaction to WWII. People on the west coast thought they were next, after Pearl Harbor. San Pedro is full of old military bases, tunnels and huge canon batteries. The people on the west coast fully expected to be attacked, it just never materialized. I think part of the reason he left was to put some distance between himself and the coast. Probably seemed like a good idea at the time. Especially if you weren't real keen on putting on a uniform.
Anyway, in another poem in War All The Time he says that he spent 7 years on the Philadelphia barstool, which we know is impossible. Not that it matters. He was a poet, not a journalist. ;)