Ok - the first thing to do to see if anything is damaged is to solder back in the old switches. If it works, you're still alive.
Next, use your multimeter to see whether pressing the switch makes or breaks the circuits. You might have a setup where one of the signals is connected when you push the button and one that is disconnected. When you've got that sorted, either the double pole switch that av8vfr suggested or Virago's relay would work, though it's probably easier to use the switch.
Key thing is to make sure that it replicates the make or break conditions of the original switches. Generally with those switches, the terminal in the middle is the common one, one of the terminals on the outside will be normally open, and the other will be normally closed. So, one wire to the centre, and the other wire to the correct outside pin depending on whether that pair need to be closed or open when you push the switch.