Differential amplifier and different ground
The grounds are floating.
There are two problems with the designs:

The design will be susceptible to any kind of noise, because the GND2 is floating with respect to GND1. EMI could cause GND2 to be a higher voltage (and the second board will function like an antenna)
There is no return path for the current flowing from the board with GND1 to the board with GND2 (in the diagrams in the OP). This means the only return path for current will be through the air and the current will be very very small, not enough to overcome the input bias current needed almost all amplifiers.

You'll need some kind of connection between grounds. If you want to isolate the grounds between boards use an isolation amplifier such as the adum3190. These amplifiers use chopping to pass the analog signal through an isolation transformer, then reconstruct it. (you still need to provide a ground from one board to the next, but the grounds are not tied together on the same board). If your trying to save on wiring, there is simply no way to pass an analog signal between boards without a return path for the current (it's like trying to get current from a battery without connecting a wire to the other terminal).

Source: https://www.analog.com/en/products/adum3190.html

will the below circuit work provided the output voltage of the first
op amp is within the voltage range of the second op amp?

There will be no reference for the ADC to base the voltage because GND1 is floating, it will pick up a reading but it would be similar to connecting a wire to the ADC and air.

For the circuit shown above, the ADC will measure the output with
respect to which ground? Isn't the input to the ADC floating?

The ADC is always referenced to it's ground GND2 (many ADC's have built in voltage references that are referenced to ground).

will the below circuit work provided the output voltage of the first
op amp is within the voltage range of the second op amp?

Probably not, it will depend on the voltage of GND2 and if GND2 is only connected by air, then no current will flow between boards and there will be no way for a voltage to be determined by the buffer amplifier.