HELP!
In our upstairs bathroom, we have a sink, and a stand up shower.
This afternoon, only the hot water would run from the sink faucet, but nothing would come out in the shower.

We wondered if the pipes were frozen because of the cold weather. This evening, we tried the tap and showers again. We discovered that the hot faucet would give us hot water if we ran it for a while (but it was ice cold when it first came out), cold water would still not come out. When we turned on the shower at the same time as the hot water was coming out of the taps at the sink, the water started to come out in a mid-to low trickle. As soon as we turned off the sink faucet, the shower would stop giving us water as well.

As a last observation, there is a fireplace in the next room, sharing the wall with the plumbing - the flu was left open a night or two, and the floor is VERY cold in the room. My first thought was that the cold air coming down the chimney has chilled the pipes enough to freeze the cold water...

Is this likely a cold weather problem, connected to ice? Or could it be a different plumbing issue?

I am pretty sure that this is a weather problem. I think you have one or more frozen pipes. The oddity of turning on you tap and the water coming our the shower as a trickle is because the water in the tap builds up enough pressure to push water out of the shower. If you turn on both the hot and cold at the bathroom sink and hold a rag over the faucet to keep the water from coming out it will make a larger stream at the shower.
The thing you need to watch for now is the fact that the pipes could have burst when the froze and as they thaw they make leak water into places you don't want water.

I've had at least 4 complaints today about water pipes freezing. Consider heat taping your exposed pipes. Click on; Heat Tape Product Info and Reviews at DoItYourself.com to check it out. Good luck, Tom