Hard to start in cold weather, now saying “transmission fault” 2008 LR3

Hello, slow start and "transmission fault" do not sound like they correlate as a singular cause of your troubles. You may have more than one thing going on here. That said, if you have low battery voltage overall, you may negatively influence transmission electronic operation. Start by checking your starting and charging system -- by checking your battery first -- and then go from there. Once you resolve the hard start, test for the transmission concern. With a warning indicator like that, you will have a diagnostic trouble code (or more than one) that will give you clues on where to look next.