Do I definitely not want to to add a ground wire to this UART connection?
There's no definite answer for this, specially without knowing what it is.

If you don't add the GND connection:


Return current of UART signals will go through the "Y" cable. Huge loop, possibility to radiate or suffer from EMI.
A high current or high frequency in a branch of the "Y" cable could make the grounds of the boards to be on a different voltage, possible errors in transmission


If you add it:


Possibility for a ground loop, which could cause EMI problems
You might make a stronger ground than the "Y" and some high current or high frequency could flow there distorting the signal.


Without knowing what it is, I recommend adding the ground wire to the UART interface and also adding a strong ground connection between the boards. The strong connection will carry all the high currents and high frequencies. The "UART GND" will only carry the return currents of the UART signals as it is supposed to.

Be warned, my guess on what your setup is could be wrong and so could be my answer. If you give more details on what the boards are, you can get a more detailed answer.