Can you post up a pic of your switch?
It shouldn't have a nut to attach it to the bottom, it screws in just like a door jamb switch, see Badsport's pic.
The switch makes the ground, so it needs to thread into the tab, otherwise you could have a problem there.
How did you test the bulb socket? With a regular continuity tester with a light? If so, you should be good on that end. This circuit is hot all the time, so when you grounded your circuit tester you bypassed the switch and provided the needed ground rendering the switch "useless" in this test. Are you sure you have the right bulb, and that it is a good bulb?
Try using your circuit tester as a ground with the switch. Connect the test light to ground, holding the switch in your hand, when the probe end is set against the side of the switch the light should come on with the switch "open" and go off when you "close" the switch.
Try this and report back.
C