Progress...
Checking the cache files, the localStorage data is retained, but is just not read by the webview on restart.
Solution (I hope)....
There is a second webview in this application, it is only used to display logs and so not linked to WebViewExtras and WebViewSettings.
However, applying the same WebViewSettings that are used on the first webview (that was accessing localStorage) onto the second webview, appears to allow the main webview to read localStorage.