Hi!
I have the same problem a lot of the time, and have gotten into the habit of hand-washing the dishes shortly before first practice session of the day. (I tend to doze off at night with a pileup in the sink, so shoot me.) If I am noticeably stiff from cold or from overzealous practice the previous day, I don't play until I'm better, but that's not frequently, thank de lawd. One thing I do in cold weather that helps right away is I start out wearing long (mid-forearm), soft-cotton-knit fingerless gloves--one hole for the thumb and another large hole for all the fingers. This can be an issue if I need to be playing a passage with hands crossed close to each other, but so far I've never ended up with my right thumb in the left fingerhole or anything like that! I think the reason this helps even though the fingers remain uncovered is that there are no muscles in the fingers, only tendons, so the kind of blood circulation you need to overcome the "cold" (or stiffness from cold or any other reason) is aided by merely covering the body of the hand, the wrist, and lower forearm.
Cheerio!