This isn't really a historical question; it's a science question. But the main issues that get in the way of observing an atomic bomb at that distance are:

1. Altitude. If you're not really high up, buildings, mountains, even the curvature of the Earth, can get in the way.

2. Clarity of the atmosphere. Dust, humidity, smoke, whatever, makes it so that you can't see infinitely far through the atmosphere.

If you were really high up and didn't have to worry about all of the atmospheric humidity? You could see an atomic bomb going off on Earth from the Moon. (It'd look like a little spark of light.) Hence satellites are used to make sure nobody sets off an atmospheric atomic bomb without people knowing.

But in the real world, under real-world conditions? The best you can usually go from a terrestrial vantage point is 40 miles or so visibility, and that's after being very high up (e.g., at the height of about 100 stories, which is the observation deck from the top of the new WTC building). In an airplane, however, your visibility can be much further.

They are, however, very bright, and so people report seeing their flash (in the same way you can see a lightning bolt light up the whole sky, even if you don't see the bolt itself) at a distance of that far — certainly such things were reported during the years of atmospheric testing. And it was possible to see the flashes and sometimes even the huge clouds at the Nevada Site from Las Vegas, which is 90 miles away.

I haven't seen the show, so I don't know if what they've shown is realistic. But 200 miles is pretty far, and if it's a World War II atomic bomb, those are probably not large enough to be seen over the horizon (except, again, as a flash or diffuse brightness). Multimegaton bombs like those developed in the 1950s are more plausible (their clouds reach up much higher, and their overall brightness is much higher). I'm inclined to suspect it is not that realistic from your description.