Chromium on Android does not have automatic updates like Chrome. This app automatically installs Chromium and keeps it updated.
If your device is rooted, you can install this app and forget it. If root is not available, it will download updates but you will have to confirm installation.
Disclaimer: This app is not in any way affiliated with Google or the Chromium project.
Source code: https://github.com/adolfintel/chromiumUpdater
License: GNU GPLv3