When I enter my GOG Galaxy and click on the play button in the witcher 3 tab nothing happens, it updates the time since I last played it on the left of the screen, but other than that nothing.
Is the game running in the background (task manager)?
Q.: When I start the game nothing happens, but I can see the application running in the background.
A.: There has been a solution circling around that worked for quite a few people, thanks to Hycinthus:
1. Uninstall Galaxy
2. Open command line (Start->type "cmd" -> right click "cmd.exe" and click "run as administrator")
3. Type sc delete GalaxyClientService
sc delete GalaxyCommunication
4. Go to //gamedir\bin\x64 and rename or delete galaxy.dll
5. Launch the game, an error message will pop up, but the next time you start it everything should be fine.
From Witcher 3 FAQ.
If this doesn't help: Did you change something since the last time you played Witcher 3? (e.g. upgrading to Windows 10 - if this is the case install Visual C++ Redistributable for Visual Studio 2012 Update 4 (64 bit))