The issue was due to the following definition added in the wp-config.php file.
Edit wp-config.php and find the below line
define( 'DISALLOW_UNFILTERED_HTML', true );
And comment the above line. So now it looks like:
//define( 'DISALLOW_UNFILTERED_HTML', true );
You can now be able to edit the email templates and save.
Once you have completed the editing, please uncomment the definition. (Just remove the two forward slashes)