Thanks to you and to parna.
On the path to a cure I shut down the computer and waited until all settled then restarted the computer.
When the sign in window appeared I did so, but then held down the option key which eventually took me to the window of choices.
The amount of broadband allowance used to upgrade both computers and all the relevant applications required by Mountain Lion plus having to redo the upgrade of all the relevant application upgrades due to the AppStore 'timing out' saw over 17GB used,
This hampered (prevented) a reload of Mountain Lion.
I had already tried three (3) shut downs and restarts to no avail.
However, going back to the window showing choices, I chose to restart using OS X 10.8. This was a different way of achieving the restart,
Having said all of that, everything now appears to be as it ought to be, and I thank you for assisting with ways of overcoming the problem.