I believe you misunderstand the cause of a nuclear winter.
The cause isn't the nuclear radiation per se. It is the extreme destruction of many large urban centres simultaneously. The problem is that all of those cities burning simultaneously would send up a tremendous amount of ash and smoke that would block the sky for quite a while.
This would cause a reduction in the amount of sunlight that would be available for growing crops and combined with the almost total destruction of global economy could result in the destruction of the entire human race through mass starvation.