PS edits not updating in LR

Can you please clarify what you mean by "saved as a copy". Any additional versions that are made solely in PS, will not be automatically seen or imported by LrC.

To summarise normal process: starting image sent to PS using "edit a copy with Lightroom adjustments". In the case of a Raw, that will be the only possibility so no other options are presented. Image now seen in PS shows effects of prior Lightroom editing. Further editing applied inside PS. 

If LrC and PS are version intercompatible, that image seen inside PS initially is not yet saved to disk or imported back to LrC, and its document title temporarily repeats the filename of the underlying source file you started with (even though you do not literally have the stated file open In PS). In this case clicking Save will create a new file on disk, import this to Catalog, and update PS document title accordingly. Further Save will do a further update of what is on disk and of what is seen in LrC.

If there is not that intercompatibility OR if you've used an additional external editing preset, the new file on disk is created straight away by LrC and that is what is then opened into PS for any further editing. That is similar to how, once an external edited file shows in LrC, sending that to PS AGAIN tells PS to re-open the same file on disk for further change. In either case, Save causes an update of what already exists on disk, and an update of a PS edit version which is already seen in the Catalog. 

But Save As will then generate a different, separate file anew. LrC will be simply unaware of. this: it will be none of its business.