I'm not great with time, but I'm not terrible either. I usually know when I'm running late (which is frequent) and I'll have a general idea of the actual time to within 15-20 minutes or so, but nothing special. If I know I have to wake up for something or I'm worried that my alarm won't go off or something, I will spend half the night waking up to make sure I'm not late and then get all sorts of frustrated with myself for not being more coherent and awake doing whatever it was that I had to wake up for. Tis a bad cycle.
As for wearing a watch, that is something that I usually do, though that has more to do with needing either a light or something to count seconds with, especially when doing floor nursing. Sometimes being precise is required, and a watch with a second hand is Very Necessary for that, especially on night shifts when that's most likely your only easy to find way to keep track of time. Of course, all that only matters when I can /find/ my watch. Wonder where it is right now...?