Normally a bad fan clutch will cause overheating in city driving more than in open highway driving. Also, if the overheating didn't occur prior to water pump failure then your fan clutch is likely OK.
The overheating might be due to an underfilled cooling system. Idle the engine for 7-10 minutes with the radiator cap off and the dashboard heater control set to hot until the thermostat opens and you can see coolant moving (some coolant may bubble out when you do this so have more on hand). Also using pure antifreeze could cause overheating. Use a 50/50 mix of antifreeze and water (or the premixed 50/50 Prestone)
If you plan to keep the car for a few years or more you should use only genuine Toyota Red long life antifreeze and mix it 50/50 with distilled water.
If you still overheat after doing the above then either your new water pump might be the wrong one or you might have a cylinder head gasket leak, but since you were careful not to drive with the temp gauge in the red zone, your head gasket should still be OK