Nuclear Winter relies on two things:
A lot of nukes going off in several different places around the world.
And most importantly, those nuclear bombs dispersing massive amounts of small particles high up into the atmosphere, where they will stay for years or decades, reflecting sunlight away from the Earth.
Now a Nuke over the North Pole isn't going to disperse anything except water vapour. I believe the ice cover over most of the South pole is a couple of kilometers thick, so Nukes there are only going to disperse water vapour as well, unless the Nukes are detonated over some areas of rocky shoreline, or (comparatively small) areas of rocky mountain.
I really don't know if pure water vapour will stay in the upper atmosphere the same way as "vapourised" buildings, soil, infrastructure, people, etc. I don't know if that water vapour would form enveloping cloud cover that hangs around for decades and reflect sunlight away in the same way, if there weren't any tiny solids for that water vapour to precipitate around.
What would be sure would be that there is less polar ice reflecting sunlight away, and more dark ocean water absorbing sunlight and warmth. I don't know which effect would be strongest: the extra warmth from more open water and more exposed soil in the South Pole, the cooling effect (only in the North!) of the saline mix in the sea-water being disturbed, or the possible cooling effect of extra water vapour in the upper atmosphere.
