Neirin's got it, use a state cell definitely. Here's a mockup:
As easy as it gets; you can add a pulse former (shown on the right) to make the lever do work on the downswing, too.
Here's one that will run forever (turns on after X seconds, stays on for Y seconds)
You can flip the lever to act as an emergency shutoff, too.
If you space out the timer/state cell and put the input from the lever in between the two, flipping the lever off will startup the tesla coil immediately.
Disregard the piece of wire leading back into the timer. I was going to get fancier with it but decided it was unnecessary.