I started a thread here
with a similar issue recently, reporting no cabin or screen heat for the first 9 miles on a cold day, and then experienced a serious over-heating incident on the next drive. The car is now off the road with a diagnosed head gasket failure and blocked radiator. In my opinion your concern is sensible.
It's been difficult to establish exactly what exactly happened to our 2000 car, but the best ideas from the guys at the Honda garage and the mechanic who recovered the car are:
1. Poor coolant condition allowed partial freezing and head gasket damage, leading to loss of coolant into engine and drop in coolant level, leading to thermostat and cabin/screen heater problems. Radiator also end of life.
2. Blocked radiator in cold weather preventing circulation, leading to thermostat and cabin/screen heater not operating (when cold) followed by localized boiling and head gasket failure (when hot).
So I suggest checking the coolant level/condition. If it is low (and no leaks) then you might have #1. If you get the radiator checked/flushed then you can rule out #2. In our case the heater and water pump are thought to be working fine.
Hope that gives you some more ideas!