It could be the first signs of exhma- for some people with exhma the cold weather can aggrovate it.
I would recomend E45 cream apply before you go out to hands and prehaps wear gloves if you are happy to do so.
Your feet may be to cold if you wore open shoes anyway- it may be better to wrap up warm and if that cream dont work see a doctor.
Also Olatum in bath can help itchy sore skin.
If you break out in hives wheni t's cold, you could have cold induced uriticaria. A doctor can diagnosis this pretty easily by applying an icecube to your skin.
I've heard other people describe this, and the only thing that tends to work for MOST people is to simply not let them get cold in the first place. This means nice warm gloves and boots with warm socks. Bring a change of socks with you to work (heck, I do this because I don't like wet socks!), and if it's not against policy a nice warm pair of slippers.