If node-webkit doesn't come bundled with the files, it's effectively just the Google Chrome browser. So it seems to me, why not just get the user to install Google Chrome, which auto-updates itself as well?
It's still standard practice to re-download a desktop app for updates. We do this with Construct 2 itself. You can make an AJAX request to see if there's a new version available, and prompt the user to download it.