The US Atomic Energy Commission (who ran the tests) did not always give advance notice of specific test dates/times (sometimes they didn't know them, sometimes they wanted them to be deliberately vague), but it definitely kept people aware of when general testing was taking place. The tests were largely one-off explosions, not multiple ones. I don't know of any Las Vegas residents thinking a war had started; it would be an odd way to start a war, with a single, relatively low-yield weapon some 60 miles off in the desert.