Basically the issue is that SharePoint uses neither the Locale of the Server or the Locale of the User but instead uses the SharePoint culture to determine the format that the Date/Time pickers will use.
In order to fix the issue you need to be able to administer the site in order to update the Regional Settings.
Go to your SharePoint website;
|"Site Actions" on a SharePoint site|
|Site Settings in SharePoint|
|Site Settings > Regional Settings in SharePoint|
Date/Time pickers will now be in UK-format (or whatever format you have selected);
|SSRS Date/Time Pickers Showing UK-format|