I am having the same issue on Windows 10 Pro 10.0.15063
Downloaded https://download.whonix.org/windows/Whonix-Installer.exe. Verified then started the installer. No installation errors or warnings were displayed during install. Installation resulted in two Windows Desktop shortcuts - "Oracle VM VirtualBox" and "Whonix for Windows". Virtualbox appears to be installed in Program Files(x86)\Whonix\Whonix for Windows.
Double clicking on the Whonix for Windows shortcut launches a simple Whonix control window with two clickable areas. On top, "Start Whonix", and below "Advanced". Clicking Start Whonix displays a brief spinning progress wheel and the text changes to "Stop Whonix". From there nothing else happens. Clicking on the "Advanced" area produces the same brief spinning progress wheel and no other visible activity. Task Manager shows Whonix-UI as a running task. There is no task listed for VirtualBox.
Double Clicking on Oracle VM VirtualBox doesn't appear to do anything. Starting it from the Start Menu does nothing.
I hope someone can please advise. Given this appears to be the identical issue to the OP maybe there's some incompatibility between the installation and this Windows build.
SOLUTION: Downloaded and installed Virtualbox from virtualbox.org. Once installed, double-clicking the Whonix for Windows shortcut on the desktop and clicking "Start Whonix" properly started everything. There must be a bug in the way the Whonix for Windows installer installs Virtualbox. Whonix team: Please let us know if what I did is okay and how to get the installer to install properly. Thx.