A rush release of the JCE Admin Component 1.5.0 beta 3 to include an option to perform an Installation Fix - manually create the necessary database tables not created by the Admin Component installation. I'm not quite sure why the problem is occuring in the first place, if anyone can she some light on this, the first rounds on me!
I also added a Control Panel admin menu link.
After installing the Component using the Joomla! Installer, got to Components->JCE Administration->Control Panel. If the dtabase tables were not created during the initial install, you should see the image below:
Click the 'Fix Installation' button to attempt a reinstallation of the database tables. Result messages will be displayed on success and failure.