Clearly the Japanese believed that if they did not surrender another atomic bomb would be used on them.
However how soon did they believe this would happen?
And how many atomic bombs did they believe the USA would have access to?
I think you are formulating the debate in the wrong terms.
There were Japanese who correctly believed that the war was lost, nukes or no nukes.
There also were Japanese, who believed that an honorable settlement was still possible, through some far fetched pipe dream scheme like Soviet mediation or Kamikaze pilot wild successes.
Nukes gave the former a decisive argument against the latter. It did not really matter how many more bombs the US had. What did matter was that the latter group of Japanese leaders could no longer deny that their near future involved a complete extermination.
The Japanese had no real idea how many atomic bombs might be available. They had not had any specific knowledge of the Manhattan Project. Their scientists were aware of the possibility that such bombs could be built, and they had a small atomic weapons research project (it didn't get anywhere).
Once the first bomb was dropped, their scientists realised what had happened within about two days, although explaining it to the military and politicians was harder.
Using the second bomb relatively quickly may have been intended to give the impression that there were plenty more. Another would have been ready for August 19th, with three more in September and three in October.
Richard B Frank's Downfall: The End of the Imperial Japanese Empire is a pretty good source on this period.
Wikipedia has some information on that. First of all, since the Japanese did have their own nuclear project, they knew that separating uranium was slow and costly, and (correctly) understood that the US would only have a few bombs. In fact, second bomb was dropped shortly after the first partly with a specific goal of making the impression that the US had a large supply.
However, on August 8th, a US pilot Marcus McDilda was captured and, under torture, claimed that the US had 100 bombs and is intending to use them soon (in reality, he knew nothing relevant). Apparently, this was taken seriously by the Japanese, at least the war minister Anami informed the cabinet, influencing the surrender discussions.