I know the popular concensus would be that Hermann Göring would succeed Hitler as Führer...
If "the plot succeeds", that includes the coup d'etat to overthrow the Nazis, so Goering gets the chop.
If "the bomb kills Hitler, but the coup fails" that's different.
Being that Hitler was Nazi Germany...
Pretty much, and IMO if Hitler was killed, the regime would collapse, despite the poor planning for the coup.
I can't see the war going on much longer
The war ends quickly - probably within three months if the coup succeeds, within six months if not. Once there are non-Nazis in power, and they realize they can't hold out for better terms, they surrender.
Göring wanted the war to be over with ASAP, so that would be another reason to surrender.
Goering has no chance of negotiating a peace - he is doomed to execution as much as any other Nazi. However he would not last; there would be a coup later on for that very reason.
And what about the effect it would have on Japan? The reason why I brought that up was because The United States was still fighting over on The Pacific, so with Germany out of the way, they US can possibly drop the atomic bomb one year earlier, thus ending the war alltogether.
The Bomb was not ready until summer 1945. Earlier German surrender doesn't affect that. In fact, it may abort the development of the Bomb. Many of the leading scientists disliked the idea of the Bomb, and worked on it
only because Nazi Germany might get it first. If Germany surrenders, and there is more than six months work left, there could be a mutiny.
And lastly, would FDR finish his tenure as President alive and not run for a third term. I know the only reason he ran for a third and fourth term was because the US needed him to get them through the war because of his superior leadership skills and massive popularity.
That was
his rationale. By July 1944, he had already been nominated for a fourth term, and would not have withdrawn. His ego would not permit it. He saw the postwar settlement as just as crucial as the actual war, anyway.
Getting back to your original question:
The Schwarze Kappelle seizes power. All the leading Nazis are summarily executed. However the Allies spurn all offers to make peace by the Beck regime, reiterating the demand for unconditional surrender. The neo-Germans have seriously unrealistic expectations anyway.
Meanwhile - the neo-Germans order strategic withdrawals on every front, IMO. No Mortain attack in France - instead everybody heads back toward Germany; they may stand on the Seine. This starts even before COBRA. German forces evacuate Italy - the neo-Germans owe Mussolini nothing. Germans evacuate Finland, Estonia, Courland, and possibly Norway. Germans evacuate Greece, Bulgaria, Albania.
Finland, Romania, Hungary, and Bulgaria all surrender and change sides, as they did in OTL August and September, but much faster (and Hungary succeeds). Slovakia too (as in OTL, but they don't get crushed by the Germans, who fall back to Moravia). The Germans evacuate Yugoslavia too.
This has two effects. OT1H, Germany avoids OTL's enormous losses and consolidates its forces for a more effective defense; and even with those losses the last year of the war was very hard fighting. OTOH, the Allies don't have to fight for a lot of places they had to fight for OTL.
It's possible that after banging their heads against the consolidated German forces, the Allies may become willing to offer conditions informally.
However, even with a more rational strategy, Germany is doomed. The neo-Germans may realize that fighting on will gain Germany nothing, while taking more casualties and bomber damage. The question is how long till they suck it up and surrender.
Germany surrenders and is divided, about as OTL. A lot of death and destruction
is avoided. Dolchstoss talk is IMO minimal. By 1944, it was fairly clear to Germans that they
couldn't win. If the Allies make even a small accomodation, then most Germans will be relieved that the war is over.
Other effects: the neo-Germans may offer to release all concentration camp and POW camp inmates to the Allies. Also, the neo-Germans may surrender Warsaw to the Home Armee. If they're clever they see this driving a wedge into the Allies, as Stalin won't be happy.
Japan: remains demented. The US and UK can shift forces to the Pacific. Perhaps more important is that the USSR will probably do so too. The Soviets probably attack Japan in March 1945, and overrun all north China and Korea by the time the Bomb comes down. The question is whether that will be enough to get Japan to surrender before the Bomb drops.
In a worst-case scenario, the Manhattan Project halts in October 1944. The Soviets overrun Japan's mainland empire, while the US and UK fight an incredibly bloody battle for the Home Islands and East Indies. (This includes the general massacre of Allied POWs and internees, something the Japanese actually had planned at the time).