Not to question anyone but I assume you checked the obvious things like belt, fan clutch, and electric fan?
Usually a gradual heat up from cold to hot with no drops or spikes is a result of an obstruction or lack of circulation. The heater system comes directly off the pump. The heater core is the highest point in the system for a reason. First it is not essential for engine operation and second most drivers will notice lack of heat before they notice the system is low on coolant. My guess is the system is low and interior heat is the symptom. As is an overheat condition. However there may be two issues. Perhaps the engine overtemped losing coolant now no heat.
My two cents.
Bill
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