Correct, the date that a third weapon could have been used against Japan was no later than August 20. The core was prepared by August 13, and Fat Man assemblies were already on Tinian Island. It would have required less than a week to ship the core and prepare a bomb for combat.
“There wasn’t a third bomb to be dropped.”
But apparently the assembly-line was going to be cranked up pretty
quickly. I can’t give you a citation, but I did hear that there were
plans to drop about 10 nuclear bombs to block Japanese forces from
getting through mountain passes. Thus allowing the Allies to land
relatively unoppossed on mainland Japan.
The Wiki report mentions seven nuclear weapons...and that the operation
was to start around Nov. 1, 1945.
http://en.wikipedia.org/wiki/Operation_Downfall
A third bomb would have been ready in 12 days. Tokyo was the intended target.