These are the steps for circuit debugging:
I worked on the assumption that all three references to 12v were from the same supply.
I know it is kind of misleading to have two 12V mentioned in the circuit.
There are 12V power supply(for 555 timer & cooler) and there are 12V XBox signal in the circuit. Do not confuse the two. One is a separate power supply which needs to be presented all the time, while the other is a control signal only. It can be there or not. No power is taken from the control signal.
Do not connect the "XBox 12V signal" to the cooler circuit 12V supply circuit yet.
1. Do not use a large value 330uF capacitor for testing. Use a 22 or 47uF cap for testing. The cooler does not have to be connected.
2. Check for the correct direction of the diode 1N4148. Turn the 2Mohm POT all the way to near minimum resistance.
3. The 12V supply for the 555 needs to be presented always for the circuit to work. No connection to the "12V XBox control signal" is needed for testing. Just leave it open and unconnected.
4. Power up the 555 timer circuit. The relay will energize and then drop OFF after a while.
Waits until this happens. Now the circuit is into standby mode.
5. Temporarily connects 12V via a jumper wire to the circuit input marked "12V control signal from XBox". The relay will immediately energize.
6. Remove the jumper, the relay will stay ON and then drop OFF after a short time delay
7. Increase the resistance on 2M POT and confirm that the drop out delay time interval increases.
8. Change the capacitor back to 330uF or 470uF and repeat step(5) until you have about 10 minute delay
Now the 555 timer is confirm working:
9. Power OFF the 555 circuit and connect "XBox 12V singal" to the input marked "12V control signal from XBox"