If, in August 1945, the Allies had known that Japan had developed a nuclear weapon, and a credible means of delivery to the W. Coast of the US, would Truman have authorised the use of the Hiroshima and Nagasaki bombs ?
Yes. He would have used nukes to obliterate the Jap(anese) nukes.
Never forget that Hiroshima and Nagasaki weren't perpetrated to terrorise the Japanese. They were ready to surrender already and the only two sticking points in the "unconditional" surrender were that they insisted on keeping the Emperor after the surrender and they insisted on retaining the Imperial Japanese flag as their naval ensign.
The nuke strikes were to terrorise the Russians, not the Japanese. The presumption was that Commies were too stoopid to devise a "deterrent" of their own.