In my situation, I have just got a new PC. Transferred LR catalogue and Pictures correctly. LR works fine, PS works fine . . . . . the only thing not working is when I launch PS from LR to make further edit and then click 'Save', it's not displaying the saved image in the LR folder (either in Library View or Develop View), BUT it is saving the file saved by PS in my Pictures folder on the HDD. No such issue on my old PC.
I've spent hours today going through everything.
I'm using LR Classic latest version and PS latest version.
The only difference between the two PC is that the new PC has Version 21.1.0 of PS and the old PC has Version 21.0.3.
Any suggestions would be most welcome!