Lightroom Classic: Can't see edits of images backed up and viewed on different devices

Lightroom is working as designed. Original images and backup copies of those images will always be maintained as original like they came from the camera.  To see the edited adjusted images anywhere besides in LR, you need to generate a derivative image that merges the edit adjustments with the original image.  This is what the Export process is intended for and why there is no "Save" item in the LR menu. 

I suspect the problem is the edits that aren't showing are the non-destructive metadata/develop edits that haven't been rendered to the files you're trying to view in applications outside Lightroom. Only Lightroom can read those edits and render them.

The JPEGs you exported have the edits baked into the pixels and the edits can be seen by any image viewer.

Yes, as other replies indicate here. LR's  main advantage over many other edits / catalogs is the none destructive feature. The edits are stored in a sidebar file so the original is untouched. LR uses these files to apply the edits made when an image is opened in LR. If you open the image outside LR you will only see the original as shot. If you export an image, then as noted above, this image is then embedded with those edits.
There are many introduction to LR resources available on line. Check some of these out as they are very instructive and help you grasp the concepts of LR quickly. It can seem confusing initially, especially if you have moved from Aperture. 

Thanks so far. Does that mean that I will have to export all pictures first to one of the backup-drives first, and - as soon as I have re-edited - have to export them again? this is extremely inconvenient, and, basically, normal backups are quite useless????

Normal system backups like Time Machine or similar are to protect and preserve your original image files.  The LR catalog will preserve all of your edit adjustments.  You export images that you want to view outside of LR.  These do not need to be backed up since you can always export a fresh copy if the derivative image is lost or no longer available. The Lightroom catalog file contains all of the information about your images including edit adjustments also need to be backed up as a regular part of your system backup and data retention schemeViewing images on your NAS drive has nothing to do with backups.    If you want images to view on your 50" TV screen, they need to be derivatives and these can be created and maintained using the Hard Drive Publish Service in Lightroom