The plan was always to bomb and invade. The atomic bombs were not a choice taken to avoid invasion. They were just another thing to throw at Japan.

As for what exactly would have happened if Japan hadn't surrendered unconditionally on August 14th, we can't know. That would have depended on choices yet to be made by the United States, Soviet Union, and Japan. But we know what was already in the works. The Soviets had declared war on Japan on the evening of August 9/10, and made huge inroads into Manchuria. They would have continued and cut-off Japanese supply lines in mainland Asia. The US would have likely dropped another atomic bomb within a week of August 14th — they had the next one ready to go and there were loud rumblings from people in the high echelons of American power to use another one. (Exactly where is unclear, but Tokyo was a commonly-advocated-for target.) Truman had, in the summer of 1945, approved the invasion of Kyushu, the southernmost Japanese island, beginning in November 1945. Whether further atomic bombs would have continued to have been used strategically (against cities) or tactically (to help with the invasion) was a discussion already in the works, but the US would have had been able to produce at least 3 bombs per month, potentially more depending on changes made to the bomb design.

Truman had not authorized the invasion of Honshu (the main island); he was taking a "wait and see" approach to that, pending the results of Kyushu and everything else. Whether Japan would have lasted that long is unclear; they were already on course to have massive famines, and their army had already long concluded that there was no way they could hold off both the US and the Soviet Union simultaneously.

There is a cottage-industry of people who speculate about the casualties of invasion, usually trying to come up with the largest numbers possible, as a justification for the use of the atomic bombs. I don't really find that a realistic or worthwhile endeavor. In general I always encourage people to resist the false dichotomy, though, of thinking it was "atomic bombs vs. bloody invasion," because that's not how they thought of it at the time, it's not clear those are the only possible options, and that's basically a trap the pro-atomic bomb people constructed after the war to justify using the bombs.