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The only possible cause of soreness of nipples in cold weather seems to be because of tactile response to cold temperature.The smooth muscles of nipple are very sensitive to cold and due to cold,these muscles gets contracted.It is the inflammation of these muscles which causes soreness in nipples.These smooth muscle are under the control of the autonomic nervous system.So these gets inflamed in winter.
You may take anti inflammatory oral or local medication to get relief from this soreness.