The white 2 way double pole white switch used here is by Switchcraft, purchased from Allparts UK (Part number: EP-0260-025) , these have been made to fit Fender Jaguar guitars... but will be fine in Red Special's plus they come with screws to attach the switch to the plate or pick-guard.
On the metal chassis of the switch there are 6x bent over metal tabs that hold the PCB board to the switch. these are highlighted in Yellow. With a pair of pliers these need to be eased back which will release the PCB from the metal chassis.
Remove the white plastic slider, you will see on the outside 2 lugs these lock the switch in to the chassis causing the click when you slide the switch. To de-click the switch these need to be removed. This can be done with a small needle file. You can see below in the images, the 2x lugs in Red and after removal in Green.
After this all you need to do, is reassemble putting the white slider back in, adding the pcb, and making sure the brass sliders on the PCB fit in to the plastic slider, and then re-bend the 6x tabs shown at the beginning to hold it all together.