If the engine warms up, and runs well at a steady warm temperature, then the water pump and thermostat are OK.
If the water pump and thermostat are OK, then the no-heat problem is either a) the heater core is plugged, or b) a connecting hose is plugged or there is air in the heater core, or c) the blend door in the HVAC system is not opening to allow heated air to pass into the passenger compartment.
a) is possible because the heater core has an internal coolant bypass; if the core is plugged, coolant can flow through the bypass instead of through the passages that transfer heat from the coolant into the air passing over them. Flushing in both directions may dislodge the crud.
b) is possible because there are two hose connections to the heater core. Coolant may still escape from the bleed hole if one of the heater hoses is plugged, or isn't flowing coolant, and air may remain in the core.
c) IIRC, the HVAC blend door actuator is located in the center stack down low, above and to the left of the passsenger's left ankle. Expose it and watch to see if it moves as the Climatronic's controls are changed from, say, heat to cold and back. If the actuator arm and door don't move, the most likely causes are that the arm or its connector has broken (this is old plastic we're talking about), the actuator has failed, or the Climatronic isn't sending the right signal. It's less likely, 'tho possible, that the wiring or one of the connectors between the Climatronic and the actuator has a fault.