In the "Manhattan" pilot, a character goes through a physical and the doctor checks him out with a geiger counter. He clicks what sounds like a good amount but the doctor says the levels are acceptable. Elsewhere we see them conducting tests with what looks like nowhere close to enough safety precautions.
Has there been any research that looked back at how often people who worked on the project came down with cancer? If the depiction of working conditions are correct, it sounds like it should be. Were there more safety precautions taken than depicted on TV?
I (still) haven't managed to watch the show (even though I've had a screener for ages now), so I don't know what they show there other than what you've described. But in terms of health precautions, they did know that radioactivity was a major occupational health hazard, and they did take steps to monitor worker's health. I have never heard of them inspecting people routinely with a Geiger counter. It would be a relatively blunt tool for checking on health (you would use it to check their clothes as they left the plant, though, which they did at Oak Ridge). What they usually did was to monitor urine for the presence of radioactive chemicals, like plutonium or uranium. By monitoring how these chemicals were excreted they could get a pretty good idea of the exposures.
As for whether there were adequate safety precautions for all of the experiments, though, it depends on what kinds of experiments (some were so important that really risky things were done), but in general, by modern standards many of these things would be considered very dangerous and risky indeed. Those working at Los Alamos knew what they were trying to do and why they were doing it, and were willing to put themselves at considerable risk to achieve results quickly. In the records of the Manhattan Project there is actually lots of discussion of this in the area of life insurance — whether the physicists and chemists qualified for "hazardous duty pay" in the event of their death in the same way that a soldier might.
To give an example of the looseness, this is a wonderfully terrible little passage I found from an official history of work at Los Alamos from 1946, describing an early mishap with highly enriched uranium:
A large amount of enriched uranium, surrounded by polythene, had been placed in a container to which water was being slowly admitted. The critical condition was reached sooner than expected, and before the water level could be sufficiently lowered the reaction became quite intense. No ill effects were felt by the men involved, although one lost a little of the hair on his head. The material was so radioactive for several days that experiments planned for those days had to be postponed.
We accidentally created a critical mass, and some of the worker's hair fell out, but it was no big deal. Yikes.
It should also be noted that they had some idea about the toxicity of radiation (it had been studied since the 1920s) but not great ideas. So their standards for safe exposure were several orders of magnitude higher than they would be today.
As for whether there was a higher incidence... to my knowledge there has not been any systematic study. The problem is that some 600,000 people took part in different parts of the Manhattan Project, and many of them scattered to the winds in the immediate postwar. Of the scientists involved, many were pioneers in radioactivity and particle physics and may have had dangerous exposures entirely separate from their work on the atomic bomb. They also lived in an era of extremely heavy tobacco use, which increases overall cancer risk dramatically, especially in the presence of radioactivity (there is a synergistic effect for many kinds of exposures). There is not even a census of people who worked on the project, much less their health outcomes. So we don't have enough data to say.
The two main scholarly books on this topic are Barton Hacker, The Dragon's Tail: Radiation Safety in the Manhattan Project 1942-1946 (Berkeley: University of California Press, 1987), and J. Samuel Walker, Permissible Dose: A History of Radiation Protection in the Twentieth Century (Berkeley: University of California Press, 2000).