The latest update pushed out to the Skype UWP app on Windows appeared to have broken Skype's SMS syncing, but there is no reason to worry. All it takes to get your SMS syncing up and working again is to re-enter a code (via NokiaPowerUser.)
According to the folks over at Nokia Power User, all it takes to get everything syncing again is to Launch the Skype app on your PC, go to Settings from within the app, and scroll down till you see the code. Next up, you should launch Skype on your mobile device, also go to settings, and enter in the same code that you see displayed on your PC. After that your PC will appear in the list of connected devices, but if it does not, you might want to disable SMS sync on both PC and Mobile, and then re-enter the new code.
Has this trick worked for you? If it has, please do let us know by dropping us a comment below!