It sounds like you have a coolant circulation issue like Dean said but I would not be willing to point a finger at the water pump just yet.
Fact is I can not ever remember finding a pump that did not pump water. The bearing may go out and you can shake the fan around; the seal may go out and it leaks; or pieces may berak off causing the pump to explode; but never one that ran good with no squeaking stop pumping water.
Even without a fan and or shroud and or engine timing way off the system should keep up with a idleing engine especially this time of year.
If you check your old thermostate in a pot of hot water and it opens you will have to look elsewhere for the problem.
I am betting on either (in this order)
Slushed up because of cold weather
Hoses going bad
or a piece of junk has broken off and blocking your engine water ports.