This is a common misconception. I highly recommend having a read through the book "Nuclear War Survival Skills" by Kearny. It's an older book more focused on the Cold War, but still highly relevant today. It systematically dispels this kind of myth. You can find PDF copies online or pick up a used copy cheap.

Maybe those at strike ground zero wouldn't know it happened. Those in the outskirts would begin figuring it out pretty quickly.