Most of the radiation after the blast is only a problem if ingested or inhaled and even then a short exposure is not likely to be a short term problem for a healthy adult.
The people that die of radiation sickness after an atomic bomb are usually close enough to the bast to get neutron radiation and gamma rays in fatal doses and/or are exposed to high levels of short lived radioactive isotopes in their food and water in the days after the bomb. Neither of those applied to people visiting the trinity site, they really only risked a higher rate of cancer years later.
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