Sounds like there should be a link between both of the commons, from the photo it looks like there is nothing in the second common. As your fitting a metal face plate there must be an earth in the back box or earth connected onto the face plate.
Remove plug from socket. Brown from plug goes to common in light switch. L1 Brown from switch goes to light. There should be an earth terminal in switch connect both G/Y. Use a terminal block or Waggo connector to connect both blues together.
Have you got a voltage stick to check if there is power at the isolator? The options for no power are: RCD has tripped. Lose connections, normally at the isolator. Dont try checking with power on. Turn power off. Isolator has failed. You should get an electrician to check the circuit.
