"basically the same sort of climate" is somewhat overgeneralizing it, as the mainland of Norway is quite a bit warmer than the rest, which has polar climate. So it appears that the mainland of Norway is the only landmass of the four with Norway lemmings. (Svalbard, and Jan Mayen by extension which is even a bit further to the south in latitude, are a bit tricky, as the climate isn't as harsh as on Bouvet island and Antarctica, but it seems like there are no lemmings, at least no Norway lemmings, there. I didn't include Peter I Island either as it's pretty much the same deal as Antarctica. Useless trivia: Svalbard and Jan Mayen, as well as the very remote and uninhabitated Bouvet island have their own ISO country code and subsequently even their own top level domain.)
Independent on which hemisphere they're on, Queen Maud Land and Bouvet Island don't appear too lemming friendly, with the latter covered to over 90% by ice, and having very little vegetation, and you can imagine what Antarctica is like.
Ok, go ahead with a new question. You were pretty close, and Svalbard was admittedly somewhat tricky.