Does Jr the third nuclear bomb exist?

Answer 1

No. It's entirely fictional, though the book it appears in is written in a style intended to make it appear as if it were real (it's somewhat similar to some of Michael Crichton's works - The Andromeda Strain, Jurassic Park, and Prey, for example - in that respect).

The third nuclear bomb was "Fat Man", the bomb dropped on Nagasaki.

The second and first were "Little Boy" and "The Gadget", respectively. Little Boy was dropped on Hiroshima, and the Gadget was the device used in the Trinity Test.

After Fat Man was detonated, the US had no remaining atomic bombs in stock, though some effort was made to make the Japanese government believe that they did. Obviously more could have been made (and would have been, had the Japanese government not surrendered), but it would have probably taken a few weeks.

The fourth nuclear bomb (never named) was assembled by Los Alamos and in San Francisco for shipment by air to Tinnian on August 18, 1945 (4 days after Japan surrendered, so it was returned to Los Alamos). It probably could have been dropped between August 25 and August 29, had Japan not surrendered.

The US had an additional 20 bombs in various stages of fabrication scheduled to be completed and dropped on Japan in 1945, if needed: 3 in September, 3 in October, 7 in November, and 7 in December. The increase in production rate in November was due to a planned change from pure Plutonium-239 cores to composite Plutonium-239/Uranium-235 cores.

Except for the MK-1 Little Boy gun bomb all other nuclear bombs in 1945 were MK-3 "Fat Man" style implosion bombs.

Source: Swords of Armageddon by Chuck Hanson (Classified documents obtained via FOIA).