The Correct voltage reading should be 12V (not 1V). When you connect the ground of the Arduino to the ground on the Mezzanine board, you get the correct reading. When you connect the Vcc on the INA219 to ground on the INA219 you have shorted out the power supply, and as expected it doesn’t work with no power.
The Vin+ and Vin- terminals on the INA219 ARE NOT “power” and “ground”. Vin+ should be connected to your power supply (positive), and Vin- should be connected to the load (the center pin on the 410c power connector). The “current” flowing from Vin+ to Vin- is measured by the INA219, The INA219 also measures the “voltage” between Vin- and the Gnd pin on the INA219 board. You need to connect the Gnd pin on the INA219 board to GND somewhere on the 410c or the mezzanine board. You also need to connect the Gnd pin on the INA219 to Ground on the Arduino, Note: two ground wires.
When you connected the UART adapter to the 410c, and you connected the Arduino to your host computer, you effectively connected the ground on the 410c to GND on the INA219 (the second GND wire). The ground path was from the 410c, through the UART adapter, to the host computer, then from the host computer through the USB cable, to the Arduino, then from the Arduino to the INA219 through the wiring from the Arduino to the INA219. This is a really complex connection but it works. When you disconnected the UART adapter you broke the ground connection and hence you read only 1V at the INA219. This is why you need the second ground wire from the INA219 to the 410c.
The INA219 computes the Power by multiplying voltage (in volts) by current (in Amps) the get the total power consumption (in Watts).
Full Disclosure: I am an employee of Qualcomm Canada, any opinions expressed in this or any other post may not reflect the opinions of my employer.