Let's assume that:
the entire nuclear arsenal available on earth is launched and detonated, completely obliterating civilization but leaving a few humans and animals in very remote areas alive.
this creates a 50 year long nuclear winter almost completely blocking sunlight and rendering photosynthesis impossible (I'm not sure if ~20,000 nukes would be enough to achieve this but let's pretend it is).
these remote areas have levels of radiation that are low enough for some of the survivors to not die of radiation poisoning/cancer during those 50 years
the survivors have access to enough potable water to survive 50 years
the pool of animals that survived is too small for human survivors to rely solely on hunting for 50 years
the reserves of pre-apocalyptic conserved food (canned food, for example) are too small for human survivors to rely solely on them for 50 years
there are not enough human survivors for them to rely solely on cannibalism for 50 years
Would it be possible in that environment for survivors to not starve and feed themselves throughout the 50 year long nuclear winter?
I'm not asking about complex lifeforms in general or issues with the oxygen/CO2 cycle as that would be too broad although surival of humans obviously depends on that.
I'm pretty sure some bacteria and microorganisms would survive, maybe some insects too. Mushrooms would probably survive too.
I'm not sure if humans can survive on a diet only consisting of mushrooms and cockroaches though. How do you think those poor survivors will feed themselves?