I agree that some sort of life would survive, but the end of the carbon cycle and the vast majority of terrestrial and aquatic food chains would ensure pretty much only the least complex life forms would endure. Humans could buy time with synthetic measures, but I'm not confident we could do that long-term. It might just be a long, slow decline.
In a nuclear winter situation, lack of oxygen (or abundance of carbon dioxide) wouldn't be a huge problem since we're "only" talking several years at most. A lot of plants would die, mind you. Artificial light is effective for raising some kinds of plants, but it would necessitate a lot of energy production to power those lights. Solar power is (obviously) out of the question, so that leaves fossil fuels, and maybe wind and hydro power. How much food we could reasonably produce in an artificial environment is anyone's guess. Not enough to support billions of people, though.