The question really comes down to: would Japan have likely surrendered before November 1, 1945 (the beginning of Operation Olympic, the invasion of Kyushu), without the atomic bombs? It is worth just noting that this has been debated for a long time, not just by historians, but by military analysts in the 1940s and 1950s, based on interviews with Japanese officers and generals and the like.

The basic conclusion is: it's reaaaaaally hard to say. Anyone who is supremely confident in the answer has an axe to grind.

On the "yes, they would have surrendered prior to November 1st" sign, you have people who point to the effect that the Soviet declaration of war and invasion of Manchuria had on their high command (fairly strong), the fact that the Japanese had done their own military analyses well prior to that which concluded they would be defeated by a Soviet invasion, and the fact that the blockade and conventional bombing campaigns were putting Japan on the brink of total starvation. The US Strategic Bombing Survey famously concluded that Japan would have likely surrendered before November 1st even if they hadn't been threatened with invasion — that they were just on the brink of it anyway. (You don't have to take them as authoritative; I'm just adding them to the list.)

On the "no, they wouldn't have surrendered," you have some of the hardline militarists who were totally delusional about their survival chances, you have the fact that even with two atomic bombs and the Soviet invasion, the Japanese surrender was not a foregone conclusion, and you have the argument that it's too hard to tell how important the Soviet invasion was on them without also coming right after the atomic bombs.

I would also add an additional complication, which is that the November 1st date was the (authorized by Truman) invasion of Kyushu. The invasion of Honshu was planned for early 1946, not yet authorized by Truman, and may have had other considerations — one could imagine Kyushu invaded, but then surrender occurring before an invasion of Honshu, depending on how that went, for example. And if we really wanted to get complicated, we could ask whether the surrender would have happened exactly the way it did if we dropped out the Nagasaki bombing, or if the first use of the bomb had been a casualty-free "demonstration," or other scenarios.

All of this comes down to what one thinks the causes of the Japanese surrender in August 1945 were (which are hard to disentangle because they overlap), and what role specifically one thinks the atomic bombs had in them. It is not as simple (as is often portrayed in American textbooks) as "bombs were dropped, then they surrendered." There was a lot more going on at the same time (the Soviet invasion being among the most notable things).

Hasegawa's Racing the Enemy is sort of all about this question. He largely gives weight to the Soviet invasion as important, but is not totally confident that zero atomic bombs gives you the same result. That doesn't necessarily mean the invasion happens, though; even if the war wouldn't end in early August 1945, that still leaves some time before the November 1st invasion date.