We will never know what Roosevelt's final decision would have been since he died early 1945 before Germany's ultimate surrender. But, is there any hints at FDR's opinions on nuclear weapons, what situations he deemed appropriate for their use, and if one of those situations was what the Allies saw themselves in in late 1945, a drawn out long war?
While I can't provide an in-depth answer to this question, as none of the biographies of Roosevelt I have read have gone in-depth about FDR's plans for the use of the atomic bomb, it seems probable that FDR would have authorized it. FDR authorized the creation bomb and his actions were solo, to the point that Truman was not aware of the existence of the atomic program until after FDR died. Even as VP, he had no clue about the existence of the atomic program: it was on a strict need to know basis, and in the days after FDR's death, he got a startling phone call revealing the development progress of the most potent weapon in history.
As the chairman of the Truman committee, there was at least one time where he got somewhat close to discovering the existence of the atomic bomb, as either Steve Neal covers in Harry and Ike, David McCullough in Truman, or Annie Jacobsen in Project Paperclip, and he was warned off and the scrutiny shut down. I can't remember if this was during the VP tenure of Cactus Jack Garner or Henry Wallace.
Admittedly, there could be arguments for the other side, that FDR would have backed down: while FDR comes from a patrician family, Truman does not and had, in fact, experienced combat on the front lines as a soldier in the First World War. One might easily argue that this hardened Truman enough to make the decision and that, at the decisive moment, FDR would have backed away from the button. There's no way to know for sure, but as FDR did authorize the entire automatic program and oversaw, from a distance at least, its progress, it seems unfeasible that he would be unwilling to use it. Germany likely would not have hesitated to use an atomic weapon, and the point of the Manhattan Project was, as I understand it, to beat Germany to the creation.
Sources:
Annie Jacobsen, Project Paperclip
David McCullough, Truman
Steve Neal, Harry and Ike
H.W. Brands, Franklin D. Roosevelt
Conrad Black, Franklin D. Roosevelt
We don't have a lot of insight into what FDR was planning to do with the bomb. There are very few sources on it, and he didn't write down his innermost thoughts. He died before really concrete planning took place.
A few of our scant data points:
In September 1944, FDR and Churchill signed an agreement known as the Hyde Park Aide Memoire. It includes the following line: "when a 'bomb' is finally available, it might perhaps, after mature consideration, be used against the Japanese, who should be warned that this bombardment will be repeated until they surrender"
Shortly after that, he met with Vannevar Bush, who was an administrator on the project, and asked him whether the bomb was meant to be only a threat or really a weapon. Specifically, in Bush's recollection of the meeting written only a day or so afterwards: "The conference lasted for about an hour and a half and it was of course difficult to record it in full, although I believe I recorded the essential points accurately. At one time in the conversation the President raised the question of whether this means should actually be used against the Japanese or whether it should be used only as a threat with full-scale experimentation in this country. He did so, I believe, in connection with Bohr's apparent urging that a threat be employed against Germany, which would of course, I think, be futile. I stated that there were many sides to this question, that fortunately we did not need to approach it for some time, for certainly it would be inadvisable to mare a threat unless we were distinctly in a position to follow it up if necessary, since a threat which had no effect and was not followed up would have the contrary effect to that intended, and that it seemed to me that the matter warranted very careful discussion, but this could be postponed for suite a time, and the President agreed that the matter did not now need to be discussed."
In December 1944, FDR apparently asked Leslie Groves, head of the Manhattan Project, whether they might have one ready for use against Germany in the near term. Specifically, according to Groves: "President Roosevelt was quite disturbed over the Battle of the Bulge and he asked me at that time whether I could bomb Germany as well as Japan." The answer was no.
And that's about it. So that is not much of a documentation base to go off of, for diving his thoughts. It seems like if he had the bomb in time for use against Germany, he might have been interested in that, but it would have been logistically difficult to pull off. He did not survive to the actual planning phase, though. He did not apparently preclude using it against the Japanese, but it doesn't look like he had active plans to use it. Which makes perfect sense if you remember that the whole enterprise was somewhat uncertain anyway — until the spring of 1945, the fact that an atomic bomb would be ready to use was still not clear, and so nobody was basing their plans around it. Even through the spring it was not an active factor in major war planning until around the time of the Trinity test. The fact that FDR authorized the program should not be taken as evidence that he would have approved dropping it on a city — that is a separate step (aside from the possibility of using it as a deterrent or a warning or a mere threat, there are also the possibilities of not dropping it on cities as targets, but dropping it on more isolated bases, for example).