No, the Great War in the Fallout universe happened in 2077. The 50s revival was just the fashion at that time.
Just like the 50s were a time when technological advances were spreading rapidly, the Great War happened during the time of technological breakthroughs like personal assistant robots and fusion power.
Ok, that does explain some things. But it seemed to me like the technology is all kind of slow and bulky. The mech suits, and the doc was able to outrun those turrets. I thought that was because it was the 50s when the war happened, but I guess not?