The basic timeline is:
August 6: Hiroshima is atomic bombed. Many hours later, after hearing US propaganda reports of this, the Japanese high command send a scientific team to Hiroshima to investigate.
August 8: The scientific team reports back that the propaganda reports are accurate. First estimates of casualties also come in.
Late August 8/9: The Soviet Union declares war on Japan and invades Manchuria.
Early August 9: The Japanese cabinet meets to discuss the Soviet invasion and the state of the war. They are mostly ready to accept a conditional surrender at this point. During the meeting the news of the Nagasaki atomic bombing arrives; it doesn't change the conversation at all.
August 10: Japan offers a conditional surrender (the one condition being the preservation of the Emperor). The US rejects this.
August 11-13: The US increases its conventional bombing campaign against Japan, and within the country there is a failed coup by more junior military officers.
August 14: Hirohito announces that Japan will accept unconditional surrender.
When you lay it out like this, it is less clear how important the atomic bombs were (as opposed to everything else), and certainly the importance of Nagasaki is made much smaller than the "two bombs and surrender" narrative that is popular in the US. The role of the Soviet invasion is also quite large. But even the two bombs and the Soviet invasion were not, by themselves, enough for the unconditional surrender — it took a solid rejection of that possibility by the US, continued bombing, and a failed coup for Hirohito to push it to the final, risky position that ultimately could have led to his own death. (The US Occupation later decided to let him keep his position if he renounced his divinity.)
Even this layout misses a lot of broader context, e.g., the fact that the Japanese high command were well aware that the war was not going well, and that both the "military" and "peace" wings of the cabinet were betting on continued Soviet neutrality.
In general, it is important to not simply tell the story of Japanese surrender as being just about the atomic bombs. To do so is to miss very important details, and is usually done in the service of American propaganda (because if you do imbue the bombs with the sole "war-ending" property, it makes them seem more justified — whether you think they are justified or not, one should not ignore the other things going on).
The best single volume that looks at the end of the war from the Japanese, US, and Soviet points of view is Hasegawa's Racing the Enemy, and if you search for "Hasegawa" on /r/AskHistorians you will find many posts on it, and the final weeks of World War II.