2002 Chevy Cavalier water pump

It sounds like you have a bad head gasket and the pressure is escaping through the cooling passages and probably into the cylinders. Water pumps are a pined pump impeller to pulley so if the pulley is turning the pump is turning. You can check the pump bearing by grabbing the pulley and checking for play. Look at your exhaust for white smoke that smells like anti-freeze. That is also a sign of a blown head gasket.

There could be air in the lines, but you should feel some heat. When you know the car is at operating temperature, squeeze the topp radiator hose to see if it has any pressure.
How do you know it is getting hot?
What are the coolant levels in the radiator and reserviour like?

Do this first before answering the above to try to expel any air. Run car with radiator cap off and the heater turned all the way to hottest setting and coming through the main vents. Fill the radiator and wait until operating temperature is reached before topping off radiator and reserviour. Note any frothng or bubblnig formations in coolant. Replace the radiator cap. Turn of car after 1 minute. Let it cool all the way down and repeat. The heater should blow hot air and this indicates the system is hot and flowing coolant.
Check the above questions and let me know what you find after you try to bleed system.

You have to have pressure in the sytem for it to work. It will get hot but the radiator cap needs to be on inorder to see how the cooling sytem is really working. Even if the radiator cap seal or spring is bad and won't hold pressure, it affects the systems ability to cool. The fact that the hoses and heater are cold is odd since the temp sensor is usually on the thermostat housing which is next to the top radiator hose. So the top hose should feel hot if the sensor reads 195F. Maybe the sensor is bad. Try getting a radiator cap and see what happens.

I looked through my service manual reference for your car for an hour last night to find a good diagram of the cooling system and read about how it works. It is a little bit of an unusual setup but it is different in that there is a pressure cap instead of a radiator cap. I tried to locate where the cap is, but no diagram that I found would show enough around the cap to be able to tell where it is mounted. Also, it works not only by a pressure hold using a spring and seal to keep pressure in until a certain pressure is reached that the spring is pushed to open the seal and release the pressure and then once it is relieved the cap closes again. The pressure cap you have adds another control. It uses vacuum to keep the hoses from collapsing during the point where pressure has dropped enough that the spring allows the seal to close which creates a vacuum in the sytem as gases are exiting at a quick rate and pull air with them and this collapses the hoses and keeps the sytem from cooling which starts the overheat and release pressure cycle all over again. The vacuum control is on the other side of the spring so the spring cannot close so quickly that it collapses the hoses. The vacuum makes the spring close very slowly allowing the pressure in the cooling system to equalize with atmospheric so the system coninues to work without the overheating cycle re occuring from the collapsed hoses.
Whew, did you cath that? It is a good idea, but if the spring is bad in the pressure cap the vacuum will always hold the cap open and the system will never build the pressure necassary to cool the engine. That is why I asked you to try and squeeze the hose to see if you could feel pressure in the system as it needs around 13-15 PSI to work properly.
The fact that your exhaust is smokey is a sign of a blown head gasket which keeps pressure from building up as it esapes into the cylinders. So, it will affect perfomance more and more as the problem persists. The fact that you do not have any heat in the system but the sensor says it is overheating makes me think that there is some overheating problem becasue of the lack of pressure which is either a leak, like the head gasket, in the system or the pressure cap is bad and needs replacement.
I am sorry I cannot tell you where the pressure cap is, but I can send you the diagrams I have and you could find it by orienting it with the few things around it you can see.
I think it is somewhere near the bottom hose as well as the temp sensor because the system is set up upside down from most to take advantage of the pressure cap that prevents overheat re-cycling by keeping the passages free for coolant flow.
So, check for pressure and let me know what you find. Also, let me know if you want the diagrams that I have so you can find the pressure cap.

That sounds like the seal is bad and the pressure cap needs to be replaced. It is not allowing the system to hold pressure in order to work.

Your very welcome.
Come back anytime even if you just need advice as we are here for you anytime.

Take Care.

Dr. C