If the 555 pin 3 goes high after 4 seconds, the left half of the circuit is okay.
Lift the end of the 1K resistor going to the base of the S8050 and connect it to ground.
If the LED does not light when it goes through the the 1K to ground, troubleshoot that. Either the LED is bad/open, the resistor is perhaps more than 1k or its not grounded. Work it out.
Take a new 1k resistor, connect one end to the transistor base and the other end to 12V. You should hear the relay click on. If not, go looking. Check the transistor. The relay. The wiring.
Troubleshooting is just testing one thing at a time. Find where the voltage disappears or the current doesn’t flow. Just about ever attribute of a component can be checked, somehow. Multimeters check voltages, currents, measure resistors, capacitors, diodes, transistor, etc. It’s just a just a game of finding the wrong something. It’s up to you to understand how the circuit should work so you can verify how it does or doesn’t work.