Great Dramatic License for Understandable Reasons
My BS in nuclear engineering is from the University of Arizona and my MS is from the Air Force Institute of Technology. I worked Air Force intelligence in various assignments spread over a twenty-five year career, first in communications intelligence (the infamous "NSA") and later in what was most often called measurements and signatures intelligence.
Beyond the usual fiddling with the precise historical accounting of events to make a more dramatic movie TBOTE makes its most fantastic leaps because some of the material was still highly classified in 1947 when the movie was made.
Nobody died on Tinian from radiation. The character of Matt Cochran appears to be an amalgam of several scientists some of whom never went near Los Alamos. Dr. Norman Hilberry, for instance, who was a professor emeritus of nuclear engineering at the UofA when I attended was a personal assistant to Dr. A.H. Conant at the University of Chicago in 1942. It was Dr. Hilberry who handled the control rope on the cruciform safety rod. His wife's name was Ann, as in the movie. But other aspects of Matt Cochran seem to reflect Dr. Richard Feynmann, Dr. Harry Daghlian,or Louis Slotin. Dr. Feynmann was probably the youngest physicist on the project and worked in various capacities including supervising the data processor pool. The other two gentlemen died to radiation poisoning received while experimenting on plutonium cores at Los Alamos in the months after the atomic bombs were dropped.
The most glaring dramatic license was in the size and shape of "the bomb." The images used looked nothing like either Fat Man or Little Boy. Of course, thy would not want anyone to know what the bombs looked like in 1947. Also, Little Boy was made safe by Captain Parsons removing the detonator at the rear of the bomb from the charge that would propel the U-235 slug down the barrel of the bomb to the U-235 rings at the other end. Fat Man was too complex to be made safe the same way so it left the ground armed. Within a few years the means had been developed to remove the plutonium core of production bombs and render them safe.
Most, though not all of the information about the early bombs that was classified in 1947 has since been declassified and you can see detailed models of the external parts of Fat Man and Little Boy at the National Museum of Nuclear Science and History in Albuquerque, NM.