Is this a working Nano, and all you are trying to do is upgrade the bootloader?
Yes It is a working nano, and I am trying to have Uno's bootloader running on it.
What do the LEDs on the Nano show during the process of burning the bootloader?
They show nothing, just the power is turning red.
Do you have another device such as another Nano or Uno that you can burn the bootloader to in order to test the programmer?
Yes I have another nano, and I will try to use it as the programmer and see if the problem is with th Uno itself
Have you made any changes to files in the Arduino IDE installation such as programmers.txt, platform.txt, etc. or is it stock?
No, I didn't alter any files.
What are the various things you have selected on the Tools menu?
Programmer: Arduino as ISP
Board: Arduino Uno (I choose this because I wan the optiboot installed on the target nano device.
Then I hit burnbootloader.
Uno's Com port is also selected.