Maybe some more important information: When selecting "Help | View your synced collections on the web...", I see all mobile photos in the correct collections in the web. But still not in Lightroom...
Could you try to remove the sync.lrdata file and restart Lr Desktop.
Removing the Sync.lrdata might fix whatever the issue is. Or, at least, might trigger it to occur again. So remove the Sync.lrdata, start Lr Desktop, and then wait until it either syncs or gets stuck again and create at this point a Lr Desktop diagnostic log which can hopefully provide some clues.
You can find the sync.lrdata file here:
On Mac it's at /Users/<user>/Library/Caches/Adobe/Lightroom/Sync Data/Sync.lrdata
On Win it would be at C:\Users\<user>\AppData\Local\Adobe\Lightroom\Caches\Sync Data\Sync.lrdata
Please send me the LR Desktop diagnostig log - best as a private message with a downloadable dropbox link.
You can trigger the log via LR Desktop preferences -> Lightroom Mobile and when you hold down the alt key you will notice a generate diagnostic log button
many thanks for your answer! Honestly I totally missed it. But I did what you said as I read it already in other discussions. It didn't help. Then I uninstalled all from Adobe, also using a cleaning tool offered by Adobe. And then, after reinstallation of Lightroom CC it didn't start anymore, I always got an exception message on startup. This was about 2 weeks ago and I also didn't find a solution for this, unfortunately. So I took a weekend and reinstalled all completely, including Windows. I used the situation for upgrading to a larger system SSD drive. Now all is working perfect. But it really was very time wasting and annoying!
Glad that it's working again. - Guido