This might be a newbie question but will be the best why to load a new Joomla or Wordpress in order to redesign an old outdated website without deleting it first? This will help in showing the website owner the progress on constructing the new site before we replace the old one completely.
You can copy the whole old site first, then develop it in subdomain or subfolder or your localhost or any other places. After you think your new developed site is perfect, then you can replace the old one with your new one.
For Joomla, you may try JoomlaPack to backup/copy the old site. I don't know the similar extension for wordpress though.