You should be editing and saving a copy out of PSE and in that copy the changes will be seen. LR may be able to automatically import that copy as you save it if you are starting with a raw file in LR, but probably not if you're starting with a non-raw file. If you are starting with a non-raw file (e.g. a JPG) then you should never save over the top of it in PSE because LR still needs to have the unchanged original that its settings apply to.
Without knowing a few more details about what types of files and what steps you're doing, it's hard to guess if another workflow might be better or if something is not working as expected on your computer, or if that's just the way things are.
The workflow that does work for me using Photoshop (not PSE which I don't have) is:
Import a raw file into LR.
Make adjustments in LR.
Use Edit in Photoshop from LR.
Make adjustments in PS and noting that the document title in PS uses a file extension the same as my raw file.
Do File / Save in PS.
Back in LR I see my original file with a raw-file extension and next to it and/or stacked with it the saved file as a TIF (or maybe PSD) that has just been imported into LR automatically from my Save in PS.
Yes, I used the same workflow as you for several years on a company computer with a license for PS and it worked as you said. Since retirement I purchased my own Lightroom 5 and PSE. When editing from Lightroom to PSE, Lightroom does creat a copy (editing a copy with Lightroom adjustments)" so I am working on a copy, not the original but when I was working with PS as soon as I saved those changes in the PS copy they were reflected in the copy that Lightroom had generated. That process is still working with the Nik plugins but not with PSE. I'm new to PSE, having always worked in PS and only use it for a few things like a content aware fill, but I can't get the PSE copy that was generated out of LR to reflect any changes without "saving as" to my desktop and then re-importing.
What type of file are you editing? What type of file is the copy LR is generating on the way into PSE?
What happens if you use File / Save (not Save As) out of PSE? Does the file not get saved, or does LR just not automatically resave it?
I shoot RAW and do most of my adjustments in LR. Then if I further edit in PSE or Nik, LR generates an 8 bit TIFF with ProPhoto RGB for the color space. Save or Save As both take me to the Save As dialogue box.
If you are using LR 5.7.1 and PSE13 with ACR 8.7.1 then LR should not be rendering a TIF, and instead, merely passing the LR adjustments to PSE, and then PSE uses ACR 8.7.1 to open the raw file into PSE memory.