Indicator says the car is running hot after around ten to fifteen minutes

The dash indicator says the car is running hot after around ten to fifteen minutes of either driving the car or at idle while in park. Replaced the thermostat and topped off coolant in the reservoir. The coolant remains at the fill line and showing no signs of any leaks in the engine bay or under the car. While the car running and radiator cap removed no coolant is visible/circulating in the top hose attached to the radiator running to the cap nor any coolant from the reservoir flowing to the radiator. No unusual sounds or leaks coming from the water pump while car is at idle or driving and no check engine lights are illuminated. Seems to look like no oil is in the coolant. Replacing a water pump in this model car can be rather time consuming and looking for other possible causes of the overheating before doing so. I seem to think the coolant is not circulating through the system to cool the motor or as if there is a blockage somewhere. What could be my next steps to solve this? Any suggestions? Ideas? It is my mother's car and mother's day is Sunday and would like to get her car running without overheating as she is limited on driveability as only driving to work less then half a mile away. Please help! Thank you!