[Alan in AZ]

Well if it is a few minutes (maybe 5) & not 10 then that is just how long it takes for the heater core to get fully hot which means 2 things:

If the slider in on Min temp then your water valve must be stuck open (it defaults open: so this could be a vacuum or electrical issues or it could be a mechanical/valve issue)

If you always get full heat when it warms up then your temperature loop may be OC - most likely a broken outside air temp loop (likely @ outside air temp sensor in alternator hose). However even in this case the full cold temp setting should override it - so you may have an issues with the setting motor for the heater core bypass flaps or with the full hot microswitch in the HVAC head unit sticking on...

Lots of possibilities - start with the easy ones - see if the hot water valve vacuum is on/valve closed for full cold. Then try wiring the valve closed and testing cooling performance. These will give you diagnostic pointers as to what to do next.

Alan