Is the Arduino shield adapter causing interference?

Glad you got it going! Not quite sure I can see what you've done there... did you bend the original pins out of the way?

I would have:

• cut the tracks from TX and RX with a stanley knife (if you cut in two places 1mm apart, you can peel out the 1mm bit of copper and make a nice clean break)
  
• added two wires from the Pico (B6 and B7 are the default Serial1 pins) back to TX and RX
  
  

Are all the Edison's pins Open Circuit to start off with? It's quite a nice idea stacking them as I guess you can then choose which pins are controlled from which board... you just need to be careful that the Pico doesn't end up driving the same ones as the Edison :)