How does temperature affect wireless transmission?
Temperature isn't going to matter to the antenna or the RF field it creates.  The issue is most likely with your circuit or what is powering it.  Are all components you are using rated for under 0C operation?  Some are only 0-70C, although going down to -20C is fairly common.  Don't assume.  Check all the datasheets.

The most likely problem is the battery if that's what you're using to power this unit.  Both output voltage and capacity drop with temperature for most batteries.  What kind of battery are you using?  What voltage range does your circuit need?  Are your batteries fresh or fully charged?

You may have to rethink your power supply to be able to run from a lower battery voltage than anticipated.

Added:

One other possibility occurred to me, especially since you say everything is running from a 12V battery and you are only using 8V and 3.3V.  A bad solder joint can work at some temperatures then fail at others as the metal expands or contracts.  If this is a small device, try reflowing all the solder joints and see if that changes anything.  Preferably add a tiny bit of solder to every joint, but definitely make sure the solder gets molten for a second or so at least.  Then check each joint with a jeweler's loupe and redo anything that looks suspect.

Let us know if this changes anything.