Specifically, I am asking about the first test explosion in July 1945 at the Trinity test site in New Mexico.
I know that Robert Oppenheimer died of throat cancer at age 63 (he was a chronic smoker) but what about the soldiers who witnessed the first explosion in July 1945? Did any suffer from radiation poisoning or have any diseases that could be linked to the first A bomb?
AFAIK, there were no radiation or blast injuries from the original test. All observers were 10 miles away, and it was designed to test functionality, not destructiveness like later tests. However, there were people who died from radiation during the Manhattan Project itself:
This old thread discussed the radiation exposures to those on the test site in some detail. As I pointed out in it, the key study on the matter is this 1982 report commissioned from from Carl Maag and Steve Rohrer by the DoD.
To summarise, as far as can be told at this remove, exposures were relatively small. The largest ones were amongst those approaching ground zero in the aftermath, but this involved a very small number of people.
Without doing any specific numbercrunching, the numbers are all sufficiently small that any estimate of the likely number of deaths will come out less than 1, though there will be largeish uncertainties on by how much.