Before we really get into the fate of Dr. Bruce Banner, I think we need to step back and take a look at our friend, The Gamma Ray.
What exactly is gamma radiation and what place does it have in the Marvel Universe? First off, the gamma radiation we're talking about is not your father's gamma radiation (assuming, of course, that your father is, like mine, a physicist) that does boring stuff like cell degeneration and cancer. No, the gamma radiation we're talking about is part of that staple of comics, The MacGuffin What Does Give People Powers.
We've already seen this MacGuffin show up once in the form of the cosmic rays that gave the Fantastic Four their abilities. We'll see it again when we encounter mutants and the X-Gene, the Inhumans and the Terrigen Mists, and random vats of chemicals falling off the backs of trucks onto lawyers. Given Marvel's heavy sci-fi leanings, there are hosts of minor characters who got their powers due to OSHA violations. This faded over the years as the X-Men grew in popularity - eventually you could just call the random thug of the week a mutant and didn't need some tragic backstory that involved them fleeing into a suspiciously unlocked chemical factory or irradiated beach.
Since things are still early in our reboot, I wonder if it's worthwhile unifying all the different ways people get powers in our version of the Marvel Universe? Besides color palette, what's the real difference between cosmic rays and gamma rays? They both have variable, non-repeatable effects, they both change biological matter as opposed to inert matter. So why have more than one?
So let's not. Let's posit that there is some sort of ur-particle behind things like cosmic rays, Terrigen Mists, and gamma radiation. This particle binds with biological matter, raising troubling implications regarding the existence of the soul, that we are more than carbon-based machines. It is the method of approaching this particle that produces the different effects. At its base, it's the same particle, it just depends on what technique you used to access it. The Fantastic Four, though they don't know it, came closest to the most 'pure' form of the particle, and their spot of bother getting home from their Heart of Gold-esque voyage actually bombarded the Earth in it.
So that means if Bruce Banner had run onto that bomb testing range in 1960 rather than 1962, he would have died. Peter Parker would not be remembered as 'puny' so much as 'that kid who died on the field trip.' Flint Marko would have died of cancer in prison. This also explains why there was not an army of super-powered Japanese citizens following the dropping of the atomic bombs - the door had not been opened yet.
This is not to say that our particle, let's call in the Infinite Possibility Particle because, well, it will be used for things like the Cosmic Cube and Infinity Gems later on, has not been on Earth previously. It's part of the background radiation of the universe and has been both intentionally and unintentionally collected over the years.
The Kree knew how to distill it, leading to the creation of the Terrigen Mists and the birth of the Inhumans. Later, after the Kree left Earth, the Inhumans continued to distill the stuff, not 100% clear on what they were doing, treating it more like an act of faith than science (and later still, the Skrulls would finally get their hands on that tech, leading to the creation of the first Super Skrull).
Turns out our very DNA can soak the stuff up - design flaw? - which lead to the eventual emergence of the X-gene. Still, pre-1961 people with activated mutant genes are pretty rare. The rise in the number of mutants we see through the end of the sixties and into the seventies is largely due to the Fantastic Four's return to Earth.
So now that the IPP is out there, how can it be drawn out? High energy is one way, and there's no better way to get energy in 1962 than a gamma bomb. The bomb itself was designed just to blow stuff up. It wasn't until Bruce Banner was exposed to the detonation that it was found that gamma radiation was a great way to knock loose IPPs and switch them into an active state. In that state, tinged with green, IPPs react to not only biological matter, but spiritual consciousness as well, twisting the physical form into something more reflective of the inner. There are some standard effects, of course, like increased strength, healing factor, and a green hue, but how those effects manifest varies by subject - thus Abomination's scales, Harpy's feathers, and Leader's big ole noggin.
As for Hulk himself, we don't need to do a lot of tweaking. He's been tweaked constantly over the years - grey to green, nocturnal transformations to rage bases, slave of a teenager to slave of rage, etc - so we can go with the most popular and widely known version. Bruce Banner is a scientist with some inner rage problems. After exposure (I actually typed "explosure" there which makes sense) to IPPs on the back of gamma radiation, he turns into the Hulk when he gets angry, upset, or otherwise stressed. As the Hulk, he's the strongest there is. Hulk is not a dumb brute - he's fierce and clever - although as he gets angrier (and thus stronger) he gets more savage and much less likely to listen to reason.
Banner's bomb goes off in May of 1962. The Hulk spends the first few months of his existence rampaging around the American southwest, chased by the US military. It will take some time before anyone understands him to be more than a force of destruction, and even with the Avengers speaking up for him most of the world is dubious.