I am also experiencing this problem. I purchased a new computer over the weekend. My team has recently upgraded our current project to 4.15 from 4.14. My old pc was running Win 10 Professional, this new one is is running Win 10 Home. Not sure if that makes a difference. However, I have been running through the forums all day looking for a solution. I have this problem with projects downloaded from our team source control, projects in 4.14., 4.15, new projects I created. However, at one point i was able to get the project to load if I launched engine and then selected the project. So, somehow, the version selectors is not in the install. After searching through the soultions here,
Each solution references this file
UnrealVersionSelector.exe
This file is nowhere to be found. I looked in the both Epic program locations. The launcher in the Launcher\Engine\Binaries\Win64\
and the engine folders in C:\Program Files\Epic Games\
This is also happening on laptop. Running Win 10 Edu. Macbook that has been bootcamped.