Roosevelt's main contribution was providing the project with support. This was not at all a given: the idea of pushing for a full crash nuclear-weapon production project was not at all obvious (the fact that no other nation attempted it is a nice sign of that). If Roosevelt had wavered in that support, or wavered in the increasingly large additional budget and manpower requests, the odds of the project being successful in the short period of time before the probable end of the war is very low. (The Manhattan Project still holds the "world record" for fastest nuclear production project — 2.5 years from decision to bomb.)

I do not know anything about John Garner, so I can't speak to that. But I can say, the project did appeal to certain aspects that I think of as being particularly... Rooseveltian. FDR was uncharacteristically (for his time) willing to delegate major scientific and technological decisions to new agencies of his own creation (one can ask, if FDR had not allowed Vannevar Bush to create the NRDC and OSRD, would the Manhattan Project ever have gotten created? probably not), to taking expert advice on technological issues (Bush again, but also Alexander Sachs and others), and he was for his time unusually fond of government secrecy (and thus he was willing and able to keep the thing from Congress, who probably would have killed it in its early stages). All of these qualities are somewhat unusual, and indeed Truman for one undid some of this work when the war ended (e.g., banishing the OSS without a replacement; banishing the OSRD without a replacement... only some years later were the CIA, NSF, and other "replacements" created; and Truman was never a fan of scientific experts telling him what to do). That FDR would, with a literal signature ("OK FDR") approve secret, multi-million dollar sci-fi expenditures, and then go away and let the people do their work, without micromanaging or investigating or auditing (and keeping others outside the project from interfering)... that's very unusual for any US President prior or since.

So again, I think one can make a plausible argument that without FDR, you don't end up with three atomic bombs in the summer of 1945. That doesn't mean you might not end up with anything — but you probably don't have the "success at any speed and cost" project that was necessary for that particular outcome. I think people underestimate how unlikely the specific outcome was: again, if it was delayed by a few more months, the odds of the Manhattan Project being "successful" during the war was very low, and we all know how easy it is for a large project to be off by a few months. And one need only look to other nations (the UK, Germany, the USSR, Japan) to see how difficult it was for anyone else to commit to making a bomb.

One can talk about other FDR-specific contributions (e.g., bringing the UK back into the project, which was very much an FDR-Churchill thing and something that FDR's advisors tried unsuccessfully to prevent), but I have focused on the ones that I think would have strongly changed the history had they gone differently. Obviously all of this is counterfactual and thus has limits to what we can know.

I have written a bit on FDR and the bomb here, though there is more to say about it, to be sure.

I don't know much about Garner, and any historian can help me, but I've read that more or less he was a conservative Southern Democrat. While he supported the new deal for the most part, I feel like he wouldn't keep a lot of the programs Roosevelt kept going, and for all we know the depression might get worse. He also was an isolationist at the time. Now I don't know if he'd be able to run for a third term in 1940, but if he did, we wouldn't have given money to the UK during lend lease and we'd only start producing war goods after Pearl Harbor was attacked. So a Garner presidency might lead to either the Germans lasting longer or even winning, or the Russians crushing the Germans and we end up getting continental Europe going red.