It really could be a lot of things, unfortunately. Do you have any history on the car? Why did it need a new water pump?
Some random thoughts:
1. Possible the radiator is clogged? That would take a long time to show up while idling, but appear very quickly once driving
2. Possible the fan system is bad - switch, fuse, relay, motor? That would not show up quickly on the freeway, but very quickly driving on surface streets
3. Possible the belt routing is wrong? A friend of mine once did this - took me forever to find he was spinning the water pump BACKWARDS.
4. Possible you have a bad hose or seal that is expelling coolant or sucking air in? Air in the cooling system can have a lot of effects - bad thermal management, bad flow, apparent overheating as air expands faster than water.
5. Possible you have a clogged or failed thermostat? That would take a long time to show while idling, but show up quickly while driving. Could also make bleeding the system difficult by trapping air in the radiator.
If you don't have one, I would consider buying, renting, or making a cooling system pressure tester or visiting a shop that has one. Putting 15-20psi in the system will expose most cooling system leaks. It wouldn't show you anything else, but might still be helpful.
[This message has been edited by thesameguy (edited 05-22-2018).]