Unable to save edits back to Lightroom Classic

Photoshop just saves to disk. By default the saved PSD/TIFF goes back to the same folder the raw file came from (but you can change this in prefs).

So the question is if the saved file is added to the Lr catalog or not.

First check your sort order. It may be at the end of the filmstrip.

Second - both Lr and Ps need to be open throughout the save process.