British wiring and connectors are a mystery this side of the pond. You could take it to an electrician and find out what kind of connector you need to take the feed to an outlet that you can plug the router into. I'd not cut off the plug. If the switch has an open back as shown, then it probably needs to go into a box.
Did you draw that diagram or is is the one supplied with the switch? Something is missing. 13 and 23 are IN, 14 and 24 are OUT. To complete the circuit for 120v, there should be a single pole switch between 14 and 24 and 13 and 23 could be wired straight through. Or it could be a doble pole switch that closes both circuits, which would be required for 220-240v. The electro magnet would hold the two poles closed during operation.
However, I see no indication of the electromagnet or in particular, how it gets its power. An electromagnet must have both line and neutral. If it doesn't have power, it may start, but not stay on. Which is why I wondered about the absence of the power to the electromagnet. That sounds like the problem to me is no power to the electromagnet. Is there another diagram?
Sometimes there's an additional connector or a switch that may be concealed.
The more I do, the less I accomplish.
Last edited by DesertRatTom; 04-01-2020 at 08:43 PM.