Did I break my nano?

Sweet! It's always odd to me that people use an Uno for that purpose, though it does work in a pinch.

Make the following connections:

CH340/FT232 Nano
RX TX
TX RX
VCC 5V
GND GND
DTR RST (via a 0.1 uF capacitor)

The DTR-RST connection provides the auto-reset at the start of the upload. If you don't have the capacitor or your boards don't break out the DTR pin (you can also use CTS) then you can just manually press the reset button on your Nano right when the upload starts (after compilation finishes). To be able to see when the upload starts you should turn on verbose output during upload:

  1. Select File > Preferences from the Arduino IDE menus.
  2. Check the box next to "Show verbose output during: ☐ upload".
  3. Click the OK button.

You don't need to select anything from the Tools > Programmer menu. You will be doing a standard "Upload", not an "Upload Using Programmer". The Tools > Programmer selection only matters during the "Upload Using Programmer" or "Burn Bootloader" operations. All you need to do is select the port of the adapter from the Tools > Port menu.