What causes your apocalypse? Because that will be a huge determining factor on environmental effect. Are you writing a modern-Earth sort of apocalypse, which could be caused by global climate shift, or nuclear war, or natural phenomena such as an asteroid strike or a massive super-volcano erupting? Or is it a fantasy world, where the gods or wizards or the Jolly Green Giant just lay waste to everything?
Well, I would say after the dust settles (2-3 years after), most life on earth would be gone and the ground zero of the impact sites of nuclear weapons would be heavily iradiated for years. Chernobyl recovered the flora and fauna fairly quickly after the event (5 to 10 years) I think; which affected about 1.5 square miles around the plant and forced the evacuation of Pripyet; a city of 50,000+. Of course many seeds can go dormant during the nuclear winter and might sprout once there is enough water and sunshine. Likely it would be plants like clover (can pull nitrogen from the air) to recover first, then the smaller plants. Fauna would probably be limited to birds and sea life, whose population would likely explode without the massive encroachings of humans.
I think if your culture has seeds for plants, they'd grow fine with a good amount of fertilizing (fish, feces, blood etc.) or maybe in a hydroponic setting in greenhouses. Protein would be the big thing lacking, without land animals like cattle, goats, sheep etc. you'd have issues with your population getting sufficient nutrients. If they had smaller animals (rabbits, chickens) in their bunkers, they could breed them to a point for SOME protein in their diet, but probably couldn't provide enough for everyone since space tends to be limited in a bunker and you couldn't allow for much expansion of the breeding process. Once outside of course, you could make rock enclosures for them to keep them centralized and expand once you have sufficient food to feed them AND the population base.
Life After People (history channel documentary) is a great example of what would happen after man disappears, but doesn't really cover a nuclear apocalypse to the scale you are imagining for your story. Life IS extremely pervasive and given a long enough time frame, could potentially recover fully; wtih only a dying out of some species of plant. Land animals would be the big question, which creatures (most likely bugs or rats since they can live underground and eat roots etc.) would survive and how the people could use them and what they would evolve into over time.
I'd politely suggest you do some research. There have been many such stories written; you should read them. You should read them in addition to reading recent (newer than ten years) scientific articles on the same topic.
It's okay to ask in forums, but you will get much more useful answers if you say "I have read X and Y and Z and now I have the following specific questions."
In terms of natural things, probably a lot of stray animals that were once domesticated (dogs, cats, other pets) would roam the streets of cities and stuff, looking for food with no owners to feed them. In terms of how nature would continue after a nuclear apocalypse, well I'd say it would continue as normal, except in areas of radiation. Outside Chernobyl (If you haven't heard about Chernobyl, I suggest you check it out- basically a nuclear reactor overheated and blew up, causing the evacuation of an entire city), all the trees died after the radiation got into the soil and the roots couldn't grow properly ect. So growing crops in soil with high radiation probably wouldn't work, and if by some chance it did, it would most likely be very unhealthy to eat them. If the crops were grown some way away from the site of the radiation, that's an altogether different story, but otherwise it wouldn't work. As for rain, I'm no geological expert or anything but I don't think rain patterns would be that badly effected (again, I'm no expert, just a guess). As for other things, any streams or rivers near the site of radiation would be contaminated, and the water unusable. Again back to the production of food, grass and soil would have large amounts of radiation, and therefor raising livestock wouldn't be practical either, but otherwise nature continues as normal.
Another helpful resource for you might be to read some natural history/biology type books about dinosaurs and the extinction event. A lot of the effects of the asteroid are quite similar to nuclear fall-out and could give you some details. But, obviously, plants (and mammals!) continued to live just fine.
Don't quote me on this, but I recall reading somewhere that most birdlife would get whiped out in a nuclear war, since the bombs release electromagnetic pulses and birds are sensitive to that sort of thing.
But take them all with a large grain of salt until you've read up on the science too. Every writer makes their own decision for what kind of apocalypse to use and just how much to fudge the details-- and the stories work hard so you can't tell the accurate ones or their defining facts from the dramatic license. Not that any of us have to be factual about anything, but it's awkward to not know the real facts before we decide which ones we want to use.
Something else that might be important to consider would be water currents. If bombs were dropped close enough to the sea to affect the water, currents are likely to carry the radiation, heavily affecting sea life. I read something along those lines regarding the melt down in the Fukushima reactor in Japan, where they were pumping watter into the nuclear plant to cool the reactor. If I recall correctly part of the water they used spilt into the sea and carried the radiation to several places across the pacific, though I couldn't tell you exactly how wide spread it was as I read about it quite some time ago. Might be worth a little research though.
I don't think you can discount the effects of fallout on sea life either. Fallout after 50's style city-killer bombs would be punched so high that it would pretty much fall everywhere...and given the water coverage of the earth...then again, would the same mechanism that produces limestone suck those poisons out of the biological cycle... argh, we need a radio-biologist!
Oh and after Chernobyl there were massive mammal die offs recorded, yet mammalian species still survived. Lots of mutants, but surprisingly life went on.
Depends on how big a war we're talking about. Too big and the Earth would be sterilised. Limited but still world wide, and expect massive issues. Everyone knows about nuclear winters, but few think of the direct effects of radiation and especially long term effects. Few realise that the true death toll of Chernoble is in the tens of thousands. The issue is mutations, but mutations of individual cells. Those cells that mutate, if they survive to reproduce more cells like them (i.e. immortalisation) become cancers. And by calculating the increased incidence in cancers in most of Europe estimates are that up to fifty thousand people have died from the melt down. Rates of birth defects, caused by mutations of germ cells have also gone up. And of course that was only one accident.
Now imagine that perhaps a hundred such accidents happened and they weren't accidents located in a specific spot but rather explosions that scatter the radiation in atmospheric dust far and wide. And then factor in the fact that radiation lingers, half lives etc.
The reality is that even a small nuclear war would have disasterous consequences and not just for those unfortunate enough to be hit by bombs. Radiation does not respect borders. And if it can't be dissipated by dilution over space, because those areas are full of radiation as well, then there is no safe zone.
Check this wiki page out and follow the links to some of the publications and you'll see what I mean.
I didn't mean my response to sound glib, I just sort of imagined when I said mutations that people would understand that meant cancers.
I am curious what you mean by radiation being unable to be "dissipated". Unless you posit a future where Homer Simpson clones run the entire nuclear industry worldwide, I don't picture mass meltdowns, Ala Chenobyl. That said, radioactive decay is the gift that keeps giving, but even so, when the particulate matter is toxic (and chemically carcinogenic) itself...
Horrible just doesn't cover it.
Modern nukes would tend to chuck out (relatively) short lived isotopes (not counting purposely designed dirty nukes), but the spread would be much wider than a nuclear reactor accident.
Nuclear winter is definitely the place to begin your research.
On the subject of cancers, I know humans get a lot, and so do most vertebrates, but I vaguely recall reading that certain phyla are much less susceptible to it. The context was, "these vertebrates, who have different numbers of neck vertebra to us, are much less likely to get cancer." Which is one of those weird and fascinating things I never followed up on.
PS: Coming here with a basic idea, looking for pointers on where to start researching, is completely cool.
I agree with Wordwalker. It's okay to say "do your research", but by basing it off of other stories you're taking the risk of using their fiction as fact. But then again, that's the greatest thing about fiction.... you never really know what's real and because of that, your world can exist any way you want it to.
When I said that the radiation couldn't be dissapated I was referring to the event of a nuclear war. In essence the best way to dissipate radiation is distance. The further away from a source you are the better. So in essence you want to stay away from the craters and out of the paths of radioactive dust clouds. This is possible when there are only a few explosions. But now say there are hundreds. Dissipation becomes much less easy because the further away from one crater you are the closer you are to another, and unless you are extremely lucky the winds will carry the dust everywhere. You end up with the entire world becoming significantly more radioactive, and without anywhere safe. In essence the Earth has only a finite capacity to dissipate radiation or any other pollutant because it's only a finite size.