UPDATE
I have finally found (by accident) the solution to my problem.
My use of any of the 3 updated internet browsers (Firefox, Chrome, and Internet Explorer) to access my emails on the new Telstra Mail server was blocked by my anti virus program “Kaspersky Total Security”. The program has a setting to “protect against collection of information about my activities on websites” and this setting was in the “ON” position.
I had to make an exception and disable the blocking for the Telstra.com website where my emails are located. All is working including all the features of the webmail (e.g. “New Email”, “Settings”, etc).
I find it interesting that while this protection setting was "ON", I had no difficulty in accessing my webmail on “Outlook.com” or “Gmail”.
“Telstra Mail” is the only webmail site that I know of that requires this protection setting (i.e. protect against collection of information about my activities) to be disabled. Why would that be?