I hiked ~50 miles of the LHT in the winter of 2001. When we started the trail, there was no snow. As far as I can remember we hiked in January. As we got closer to Seven Springs, it started to snow and by the time we entered the resort, we were hiking an unbroken trail through thigh deep snow. We had a group of 5 so we took turns up front. It was absolutely exhausting but I was in good shape so it was no big deal.
We slept in the Adirondack shelters each night. They were stocked well with firewood which helped keep us a little bit warmer. It was still bitter cold though. If I were to do it again, I would probably sleep in a tent. It was very windy in the shelters.
I forgot to mention, at one point you'll be hiking across the ski slopes. Keep your eyes open!