[Ditto]

Or when I wonder if the Japanese knew or suspected we didn't have a third nuke available,

From what I have read, after the two that were dropped on Japan, we did have one more available in August of 45. It was still in the US. But, more could have been made in a relatively short time.

[17th Miss Regt]

Correct. Declassified documents regarding the production timelines showed that by November, 1945, the US could produce two bombs per month.

[Campion]

Yes, the third bomb was being "buttoned up" for shipment in Los Alamos when word arrived of the Japanese surrender.

The US had the capacity for one to two "Fat Man" implosion weapons per month, and one "Little Boy" gun-type weapon every 4 months or so. This held true until sometime in 1946 or 47, when "Wigner energy" issues reduced plutonium production in Hanford (until the problem was solved).