From what I know about the world of Dredd, the Earth is pretty much either irradiated wasteland or cities. There are still billions of people living in the megacities taking up all the oxygen, and there are no plants to produce more. It seems eventually the planet would simply run out or dip below sustainable levels. Was this ever addressed?
The out-of-universe answer is the Dredd universe is a satire of an overpopulated, over-urbanized police state and not intended to make scientific sense.
That said, just how long could billions of people keep breathing if the Earth's oxygen cycle broke down? A long time. The Earth's atmosphere is really, really, really, really big compared to puny humans (though not their machines).
A single human uses about 550 liters of pure oxygen per day. Not air, oxygen. At an atmospheric density of about 1.2 g/L that's about 660 g or 0.66 kg per person per day or 240 kg per person per year.
The atmosphere of the Earth weighs about 5.15e18 kg. 3/4 of that is within 11km of the surface, so let's use that: 3.86e18 kg. How much of that is oxygen? Oxygen is about 21% by volume with the rest being nitrogen and 1% miscellaneous that I'll ignore. But we want the percentage of oxygen by mass.
1 liter of air weighs 1.205 g. 21% of the volume is oxygen, so it contributes
0.21 * 1.331 g/L = 0.28g. Divide them to get the mass-ratio of oxygen:
0.28g / 1.205g = 23%.
23% of 3.86e18 kg is 8.9e17 kg. That's a decent approximation of how much oxygen is in the atmosphere. Humans can survive as low as 17% oxygen, 6.6e17 kg, so they can safely consume 2.3e17 kg.
How long will it last?
2.3e17 kg / 240 kg per person per year = 9.6e14 person-years
Lacking a census for the Judge Dredd universe, I'll use 10 billion people or 1e10. They will have 9.6e4 or 96,000 years of breathing what's left in the atmosphere before there is a problem.
The bigger problem will be the build up of carbon dioxide in the atmosphere, we're far more sensitive to too much CO2 then to too little O2. I'll leave that for someone else to work out.
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 algae to survive)
And they've even tried farming in the Cursed Earth, implying that microbial life is still working, despite the radiation.