Hopefully. We currently have facilities built specifically for this purpose called seed banks. These store seeds and genetic material from plants and animals so that in the eventuality of a global disaster (i.e. nuclear war) that wiped out entire species, we will have enough material to be able to bring these species back from extinction.
However, there is also the question of whether anything would be able to survive in an environment where nuclear war had taken place. Nuclear fallout can last for many hundreds or even thousands of years. For example, the Chernobyl disaster was a nuclear accident that happened at a nuclear power plant in Ukraine in 1986. It’s thought that the area won’t be safe for humans to live in again for 20,000 years. So unless there was some part of the world where we could safely hang out for a few generations while the nuclear fallout cleared, it might be difficult for us to survive long enough to bring back the flora and fauna to the rest of the world!
Depending on how bad the war is I’d say, and probably not the same flora and fauna we know. One thing is for sure, life always finds ways of adapting, even in environments with high levels of radiation.
Years ago they found a fungus which was using the high levels of radiation inside the abandoned Chernobyl nuclear power plant as a ‘food’ to live. Animals and plants still live in that highly contaminated area too, but have growth defects and mutations which is a consequence of the radiation.
In terms of fauna, it may be possible to bring back some species, if a similar species managed to survive. Cells were recovered from a mammoth that had been extinct for almost 4,000 years, after been found frozen in 2011. There were plans to attempt to revive the species by using the cells to impregnate an elephant, though I don’t know how far their research has progressed.