There was only one nuclear bomb tested in the atmosphere in New Mexico (the Trinity test). It was on government land — a bombing range near White Sands, New Mexico. There were certainly concerns about the radiation affecting people downwind of the test. Scientists with radiation monitoring equipment were placed in nearby towns and settlements with the ability to evacuate populations if necessary. The radiation exposure from that test did not require evacuation by the radiation standards of the time. Whether such an exposure from a single test would have significant health impacts for people living downwind was not clear then (and frankly is not clear now — it is not a large amount of extra radiation, and it takes careful epidemiological work to figure out whether any given community is suffering from cancers in excess of what they would be expected to suffer without the radiation exposure).

Most US continental nuclear testing took place in Nevada. There were people exposed to the fallout. The government scientists testified that they felt the exposure levels were low-enough to not cause problems. We know that they were wrong about this today. In any event, the mindset during the Cold War was that it was acceptable to increase the risk of some people (the "downwinder" communities) if it would decrease the risk of nuclear war for the nation as a whole. Today we probably would parse that risk somewhat differently.

Most exposure of Native Americans to excess radiation came not from testing but from their work in uranium mines, which is a different story.