When I raised thermostat, the blower immediately come on without time for heat exchanger to warm up.It is set in the auto position. The furnace is a BDP (Carrier) model 394GAW060125. Any suggestions would help before I try to replace parts or call HVAC repairperson, Thanks

There should be a delay as you are well aware of. This is usually done by the use of a fan and limit control or a perset time on the circuit board if so equipped.

Have you replaced the thermostat lately? If you have it might be wired wrong or it might be defective sending the signal to the green blower wire. This will cause the blower to come on right away. Look for a wiring diagram on the inside of the blower motor door and see if you can locate the fan and limit control. If not your circuit board might control the fan.

Programmable thermostat is 10 years old you think it may need to be replaced?

Possible. I would start by unhooking all thermostat wires from board on furnace. Write down exactly were they are hooked up before you unhook them so you know how to hook them back up. After doing this, jump r and w on the furnace board with a piece of wire. This should start normal operation of heat cycle. If you jump r and w with all other wires disconnected, and your blower motor(not combustion blower) still comes on right away, then your problem is somewhere in the furnace and not in your thermostat or thermostat wiring. Let me know if this helps, and we can take the next step.