I've been twice in January. Had great weather several years ago, but was there during the record-breaking freezing days this year. DH is willing, but not eager, to go again in January.
DS's favorite park is AK, and one of the days we were there in January, there were almost no animals out at all. They simply won't bring them out of the barns if the temperatures are too low. We did go back to AK later in the week and, while it was still cold, it was warm enough for the majority of the animals to be allowed out.
The other problem with the cold was that we couldn't do any swimming or activities where we could possibly get wet. DS really loves several of the interesting pools around WDW, but swimming couldn't be a part of that trip at all. (Now, I will say, that swimming was not a problem on my previous January trip. We had enough warm days that we could really enjoy the pools on multiple occasions.)
Anyway, the biggest issue with January is that the temperature is very unpredictable. Aside from the first few days and MLK weekend, the crowd levels are great.