Despite the damage caused by the Atomic Wars it would appear that the world of Judge Dredd isn't quite as dead as might otherwise seem. The oxygen cycle isn't broken, nor is there any evidence that people have breathing difficulties beyond that which you'd expect in a world that's heavily polluted.
The Antarctic apparently hash lush jungles.
Pan-Africa still has areas of savannah. (grasslands)
The Pacific ocean is still alive with fish and octopi (whose food supply would require algaealgae to survive)
And they've even tried farming in the Cursed Earth.., implying that microbial life is still working, despite the radiation.