There appears to be two heating elements, one is in series with the fan motor. The fan motor is most likely a permanent magnet brushed DC motor which runs on a much lower voltage than the line voltage. I also see no switch so I assume the plug is the only means of power control.
Schematic I see:
simulate this circuit – Schematic created using CircuitLab
It's not a perfect schematic but you get the idea of how the motor is connected. Pay no attention to the values in my schematic. In the above schematic, the motor is L1, BR1 are the four diodes soldered to it, and R1 and R2 are the heating elements. The heater has three wires, red, black and white. Black is the common lead which appears to connect to the incoming hot lead. The red lead is wire in my schematic between R1 and BR1. We'll call that the series heater. We'll call the remaining heater attached to the white lead the main heater. The main heater white lead connects to the line neutral along with the other side of the motor.
The motor power supply is about as quick and dirty as you can get. Instead of a proper DC supply, they built a bridge rectifier out of diodes directly soldered to the motor and put it in series with the series heater. It acts as a dropping resistor to reduce the voltage across the motor. Since the dropper is part of the heater, the waste heat is used to help heat the popcorn.
First, get a volt meter across the wires attached to the diodes and not the motor terminals. Plug it is and measure the AC voltage across them. That measurement will be the RMS AC voltage across the motor which also happens to translate directly into the DC value you will need to run the motor. Why the direct conversion of values and why not multiply the AC by sqrt(2) to get the DC value? Since there is no DC filtering or smoothing, the raw rectified AC is fed into the motor. This crude DC voltage constantly varies from 0 to peak and back to 0, 120 times per second. The RMS value of that unfiltered DC is the same as its incoming AC value. So the unfiltered DC RMS voltage does the same amount of "work" as the equivalent filtered DC voltage. E.g. if you measure 12 volts AC across that motor, 12 volts from a proper DC supply or battery will give the same amount of fan power.
Next, subtract the motor AC voltage from the line voltage (120V or 230V depending on where you live) which will give you the series heater voltage.
Now measure the resistance of just the series heater by disconnecting it from the motor and putting an ohm meter across the red and white wires (make sure it's unplugged first!). Now you can find out the rest of the values using ohms law. The current of the string well call Imotheat. So Imotheat = (Vline-Vmotor)/Rheater
For you to control the three devices (fan and two heaters) separately, you need to figure out how to power the motor on DC and then use PWM control for speed. Then you need to figure out how to power that second heater at a reduced voltage since it doesn't have the motor attached anymore. Either a bucking transformer setup (auto-transformer) or resistor will work for the heater but you will have to deal with the resistor waste heat which can be many watts. The third heater which receives full voltage is the easiest and can be controlled by relay. This is not hard but you need a lot of external components to get the voltages where they need to be.
A very simple way to do this would be to put a relay or switch on the single heater and a triac dimmer circuit in series with the motor-heater circuit. You can vary the motor speed this way but the series heater will still give off some heat depending on the dimmer setting. So you don't have full heater control but it's much easier to build.