With the key in position 0 and the battery connected. Connect the negative of the multimeter to an earth directly on the chassis, and touch the positive of the multimeter to the red wire on the back of the barrel.
You should get 12v.
Touch on yellow wire & turn key to position 1. Should get 12v.
Touch on black/yellow wire and turn to positon 2. Should get 12v.
If you havn't got 12v, touch the meter on the yellow then red wire and check you have still got 12v. If you have I would say the switch is knackered. If you loose all 12v I would have thought a dead short or similar.
To check if it is the switch just use a short piece of wire and short across the terminals on the back of the switch. If you go red to black/yellow dash should light. If you go red to black/blue engine should crank. Connect red to black/yellow & black/blue and it should run.