We don't know. The situation changed very dramatically on August 10, 1945.

The strike order of July 25, 1945 essentially authorized the military to use as many atomic bombs as they saw fit, and put no limits on when or where they would be used after the first one, and indicated that if they ran out of approved targets more would be created. The generals in charge of these matters assumed that more than 2 bombs would be used and after Nagasaki some contemplated "saving up" more bombs as they were made available so that more than one could be used at a time, or they could be used "tactically" in the service of an invasion, and so on.

But on August 10, 1945, upon being informed that another bomb would be ready to use in a week, Truman issued a "halt" on all further atomic bombing — it would need his explicit permission. So the military was no longer in charge of this question.

Would Truman have authorized further atomic bombings if Japan did not surrender? I suspect so — there was pressure on him to do so, and he did seem to be leaning that way by August 14 — but we don't really know. And we don't know how/where he would have authorized their use if he did (cities? purely military targets? in the service of an invasion? etc.). We just have nothing solid to really go on for that.

The only obvious "limits" in place were on the production of the weapons themselves. The Manhattan Project infrastructure running at full-effort could produce something like 3.5 atomic bombs per month (or one every 10 days), though with some changes to the weapon design (composite bombs) they though that might be doubled. Either way it puts some hard limits on how many bombs could be used; "nuke out of existence" is just not feasible because that would require far more weapons than they were possible of creating at that time (they did not gain the capability to produce hundreds of weapons per month until the late 1940s, for example, so out of an realistic scope of the length of World War II). And even the Manhattan Project production rate is probably optimistic; they ran into production problems immediately after the war ended that required shutting down some of their production capability temporarily. Some of that was caused/allowed because the war was over, but some of that probably would have come up anyway and decreased the amount of bomb fuel they had.

This answer from earlier today works for your question

Basically, yes, they would probably make a new bomb every couple weeks and keep dropping them until they surrendered