Why does the nuclear bomb explode in space, in Batman v Superman?

If the heat is generated by the reaction, and there is no surrounding air to heat up, the heat does not "disappear" - energy and matter can't be created or destroyed. That heat has to go somewhere. The bomb, itself, is made of matter, so would that matter, superheated, disburse? It wouldn't spread like it would in the atmosphere of earth, but the heat would remain undimished/diluted in the bomb materials that get sent out. Also, how does heat get to us from the sun? Emitted as light, not visible, but infra-red radiation, heat can be sent without matter acting as the vessel.

Regardless, how the heat and energy is distributed after the explosion does not prevent the explosion from happening in the first place, since we're not talking about a combustion reaction that would require air/oxygen, but, rather, a nuclear fission reaction, or a fusion one, depending on the bomb type.

Also, were they truly far enough out in space that they were in an airless vacuum? From your post -

resulting in Doomsday and Superman falling back to the earth's surface

How is this possible? Clearly, they must be close enough to the Earth that its gravity can recapture them and pull them back to the planet. If they can be pulled back to Earth, that means there is enough gravity that they are probably still within an area where atmosphere is still retained by gravity. It is very, very thin and sparse, but some atmosphere probably still exists, given the scenario you laid out.

The troposphere, where 80% of our atmosphere is concentrated, goes up to about 7 miles above the earth. The stratosphere, the next layer, that has almost all the rest, goes to 30 miles, and it's too thin there to support winged flight, etc. For all intents and practical purposes, from our perspective, one might think of that as being it for the "working" atmosphere. However, while incredibly thin and sparse, there are more layers to the atmosphere, with the exosphere ending at about 300 miles out. I think most people traveling that far out would think of it as space, even though it's actually still within our atmosphere, along with very, very, very thin air.

Frank Heile, PhD in physics, answers on Quora:

Nuclear bombs use nuclear reactions, so no oxygen is required. The nuclear bomb is triggered by conventional explosives, be even they will work in space since they are self contained and need no oxygen from an atmosphere.

You might also ask yourself, based on the logic that space explosions require oxygen then how do stars exist? How do super novas occur? Answer: they don't require oxygen. That is only one type of combustion. There are many different forms of combustion that do not require oxygen.

The US has actually tested nuclear space explosions during the Cold War.

In fairness, you are right that the exact explosion might not occur as it did in the movie and that was likely highly exaggerated, but that doesn't change the fact that a nuclear explosion is perfectly capable of being detonated without oxygen.