Here is how I do it:
1. Make sure you build your apps using the same package ID (com.example.myapp) and increasing version numbers. (That's the numeric version "code" in the LAMW Android Project Options form, and android:versionCode in the app's manifest.
2. Build the app and let people download the APK it from your website.
3. When uploading a new version on the web site, also update the web page, or your web server's database, with the latest version number of your app.
4. In your app, include a function that calls a particular page on your site, and retrieve the latest version available.
5. If your app detects that a new version is available on the website, inform the user, and give them a button or something that opens the download page with the default browser.
6. With a couple of clicks, therefore, your users will learn that a new version is available and will be taken to the download page. With one more click they can download the latest APK, and install it on top of the current version.
7. If you try to install a new version of the app as described above, if the android:versionCode of the new app is more recent than the one already installed, the process will go smoothly. Android will also tell you that the installation will update the app without altering the existing app data.
This is how I do it for a couple of apps of mine, and it's a private working group, so I can't show you, but I can tell you that it works.