This is a known issue with Lightroom CC2015.1 update and has been tracked down as mentioned by Julie in this thread : https://forums.adobe.com/thread/1828483
The resolution for now is to downgrade it to Lightroom CC2015.
Here's what you need to do:
1. Uninstall Lightroom CC using Creative Cloud.
2. Once uninstall, go to this page and leave it open (don't download Photoshop Elements 13): https://www.adobe.com/cfusion/tdrc/index.cfm?product=photoshop_elements&loc=us#
3. Go to this page to download Lightroom 6 for OS X or Windows:
or here for anyone using Windows: http://prodesigntools.com/trials3/AdobeProducts/LTRM/6/win64/Lightroom_6_LS11.exe
4. Install the program.
5. When you install, you'll be asked whether you want the trial or you purchased a copy. Choose Install and sign in.
6. Do not update from Creative Cloud.
Were you able to install Lightroom CC2015 successfully from Prodesign?
Go under About Lightroom and share a screenshot of it whether its Lightroom CC2015 or CC2015.1
If you were able to install it successfully and it still says waiting for connection, you should be able to use Lightroom mobile sync.
Check under collections and let me know if you are able to see the sync option.
If the above procedure is done, will any pictures be lost or anything will need to be redone?
Nope. None of the Pictures or edits would be lost as its the program which you would be reinstalling, the catalog will remain intact and all the edits are stored within the catalog.