If the water pump goes out , would the radiator warning light come on?

check the radiator censor making sure that the wires for this have not come off is there any fluid on the road when its been left for a few hours normally a car loses water any way may be half a pint a month or so this is normall check hoses for splits before forking out for a new water pump

Sufficient coolant? Check the coolant reservoir first. Then, WITH THE ENGINE COOL, undo the radiator cap and check the coolant level in the radiator itself. Fill completely full as needed.

Any noises from the water pump? It may be failing, although a small leak is usually present and you say there's no leaks.

Remove the thermostat and check it by putting it in a pan of boiling water. Does it open like it should? If not, it needs to be replaced.

You CAN run an engine without a thermostat but it will warm up very slowly but that would be a good test to see of that stops the overheating.

Do you have a temp gauge in the dash? Or just a warning light? Do either indicate overheating. If they do, turn on the HEATER [not defroster] and see if the temp goes down or the light goes off. Your radiator may be partially clogged and the heater core will act as kind of a secondary radiator.

Did you change out the thermostat? I would also change out top and bottom radiator hoses if they feel soft. They should have firm sides.

the hose that goes to the water pump is usually the bottom radiator hose, the radiator hose that goes to the thermostat housing is the top radiator hose......cheers.

Check for a broken belt. If your water pump is not turning, it will cause the engine to get hot and the radiator wont steam. If a belt is broken it would also explain the no A/C condition.

Sounds like a bad water pump or radiator. Make sure the radiator fins arent clogged or inside of radiator, water pump not circulating the coolant properly will also cause overheating, old coolant will not properly dissipate heat efficiently and may need flushed. Also make sure the coolant has a 50% mix of water as the water does most of the cooling not the antifreeze. Hope this gives you some good places to start.

the problem could be as simple as a faulty radiator cap from your description the water that is expelled when it gets hot is not being drawn back as the radiator cools down
could be a leak in the reservoir hose
I am thinking that the low coolant sensor in the top of the radiator tank is getting an air space and that is what is activating the light
I don't think it was head gasket problem as you would be overheating or loosing coolant within days not weeks I would go to an accredited radiator specialist shop and have a pressure test done for leaks in the system including the water pump leaking when running
I am saying that the vacuum created as the coolant cools down is not as good as it should be to draw water back from the reservoir to keep the radiator full to the cap

Engine temperature coolant. Your running hot when this light comes on. Check your radiator fluid. If not, check your thermostat. Also could be water pump gave out.

It is not the thermostat since it only overheat on stop n go situation. Most likely cause is a bad radiator fan. Turn on the AC and see if both the radiator, and the condenser fan works. They should both be working when the AC is turned on. Lets stick to this for now. Let me know what happened. I don't want to confuse you with too many possiblity.

I agree it is the cooling fan warning light and normally if it stays on all the time in most cases it is the water pump so squeeze the upper radiator hose and if very little resistance this could be a sign your water pump is bad but before you do this check the condition of the drive belt