cannot upgrade Qt via MaintenanceTool

Something went wrong during installation of Qt 5.3.1 (various versions) with MaintenanceTool.exe, so Qt does not work afterwards:

C:\...>C:\Qt\5.3\msvc2012_opengl\bin\qmlscene.exe test.qml
This application failed to start because it could not find or load the Qt platform plugin "windows".

Reinstalling the application may fix this problem.

During installation there were many dialogs like the attached image containing both forward and back slashes so that whatever it was trying to run didn't. I hit Ignore on all of them and neglected to take so many screen shots. But when uninstalling any version with MaintenanceTool, the same thing happens. Then when re-installing, it doesn't happen any more, but those Qt versions still don't work anyway.

C:\...>C:\Qt\5.3\msvc2012_opengl\bin\qmake.exe -version
QMake version 3.0
Using Qt version 5.3.1 in /lib