You can bypass your thermostat to determine if that’s the problem.
Remove the face plate. You should see two screws with wires attached. Using a jumper wire(or even a pair of needle nose pliers) bridge the two screws together. You should hear the fan motor start. Allow to continue running until you hear the burner kick in and start producing heat. If this happens, it’s a bad thermostat.
EDIT: This should be done when the thermostat is turned to high yet the furnace doesn’t run.
Based on your narrative, the furnace works fine with warmer ambient temperature. When it doesn’t run, that’s when you want to jump the terminals.
Since I'm assuming your not replacing fuses. After a while it just works again? That points to a faulty thermostat. After repeated short cycling because it's turning on and off so many times a faulty thermostat can just stop working. It thinks is warm enough inside. Thermostats aren't expensive and they are pretty easy to replace. I'd start there.
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