From your description it appears that there might be a couple of separate problems or maybe their combination.
Chimney flue liner insulation (in case you don’t have one)
Although chimney liner insulation is not a requirement for gas or oil burning appliances (it is required for wood burning appliances), it does improve draft by keeping the flue warmer and isolating it from the outer walls of the chimney (yours is stone or brick?).
In your case, because of a low ambient temperature the heat from the metal flue might be high enough to elevate crown’s temperature (it always rises to the top) to the point that it starts causing condensation buildup on its exterior. Maybe there are some mortar cracks / gaps under the crown block that act as vents for that heated air that surrounds chimney flue.
Flue liner could be separated or damaged inside the chimney, which allows some of the exhaust flue gasses to be released to the space between the liner and chimney interior wall. The result is the same as from the first scenario; hot air rises and causes condensation around the crown.
There are openings around the furnace vent pipe at chimney wall penetration that allow warm air from your furnace room to penetrate chimney chase around the flue. I don’t think this would have enough energy to cause condensation… maybe.
You have another appliance (maybe water heater) connected to the same chimney but NOT connected to the metal flue / liner. Therefore, by discharging exhaust gasses into the space around the liner they rise to the top and cause condensation since they have no other way to go.
If none of the above than there must be some other source of heat connected to the same chimney which is responsible for condensation. Have the company who installed metal liner come back and check the liner condition inside the chimney, and chimney top section condition; gaps cracks under the crown.
Please let me know, thanks.