mini PCs, UniFi network hardware, Mac Mini, UPS backup, all in a 0 degrees Celsius server room - should I shut it down and move to a warmer place for winter?

Generally speaking, if your equipment is actively powered it will be generating heat and therefore condensation isn't an issue as it'll be a good few degrees warmer than the surroundings. Does that preclude the possibility of condensation altogether? Of course not, but it would have to be a really quick swing from cold to hot-and-humid for anything serious to form due to your equipment staying colder than the dew point.

The bigger problem I faced was equipment ending cold-bugged. I have a setup very similar to yours and some drives would not start after being too cold for too long. I insulated the cabinet and put some ESP8266-controlled fans in there. It stays well above freezing now in the winter and cools itself appropriately in the summer.

The one thing I can't control easily is humidity. A friend of mine tried running some server equipment in a damp basement and that caused a very high failure rate. It was noncondensing, but just the damp air was enough to kill a couple of components. Aside from the odd rain storm, I get nothing that high plus the insulated cabinet helps as the warm interior reduces the relative humidity (as temp goes up, so does the air's ability to carry water and hence relative humidity goes down for the same chunk of air at different temperatures).

They will be just fine at 0.

I have the same ish kind of setup in my shed and ended up making a box out of styrofoam with some ventilation holes to let them heat that up and keep them isolated from rapid changes in temperature. I have some Mac minis from 2012, a nice and a router running along quite happily in there :)

I have run a whole bunch of stuff (attic / outside) below freezing for years and surprisingly I never had an issue.