Have a reduced flame in the natural gas fireplace when the furnace or water heater runs

Can you look at the furnace during the time it is blowing cold air and see if the flame is on?

The thermostat has the ability to control the fan, but typically the thermostat sends a "turn heat on" signal to the furnace and the furnace controls are setup to turn the flame on for a X amount of time (to warm up the air) and then turn on the fan once the timer expired (so the air coming out of the vents is not cold). When the heat is turned on is the flame and fan going on at the same time?

Not sure about your furnace, but some have a couple of LEDs that are lit up and can be used to read a status or error codes. If you have the manual you could try to decipher the code while the cold air is running and see if it gives you a clue.

I would start by trying to figure out if the problem is with the thermostat or the furnace by installing a new thermostat and seeing if the problem goes away. If the problem is still there you can just return it. I had a furnace issue where it was going off randomly and would work in heat mode but not cool mode. It turned out that the thermostat was bad and sending intermittent signals.

From the Manual, one of these two solutions sounds most likely:

Verify all connections to wiring diagram in manual. Verify connections underneath pilot assembly are tight. Verify flame sense or igniter wires are not grounding out to metal chassis, pilot burner, pilot enclosure or screen if present, or any other metal object.

Or

With fixed glass assembly in place, verify that flame is engulfing flame-sensing rod on left side of pilot hood. Flamesensing rod should glow shortly after ignition. With a multimeter, verify that current in series between module and sense lead is at least 0.14 microamps. Verify correct pilot orifice is installed and gas inlet is set to pressure specifications. Polish flame-sensing rod with fine steel wool to remove any contaminants that may have accumulated on flame-sensing rod.

The fact that it occurs only when cold could mean a poor electrical connection (metal expands when heated which results in a better connection), or the thermocouple is not properly in the flame (maybe by such a small distance that the room heat allows it to get to the proper voltage).