How can I identify the fault on an Arduino Uno board that's had the smoke let out?
Don't even think about plugging in the new chip until you verify all the power supplies are correct.  You haven't shown a schematic of this board, but it is quite possible that overvoltage took out more than just the processor chip.  If the power supplies aren't right, the next chip could get fried by the bad power supply.

Use the scope to look at the voltage on every pin without the chip installed.  Not only make sure the voltage is correct, but make sure that the power voltages are reasonably clean, not oscillating wildly for example.  A voltmeter will only show the average, so it's of little use for debugging stuff like this.