Say I have a trivial container based on the ubuntu:latest. Now there is a security update and ubuntu:latest is updated in the docker repo .
You can use Watchtower to watch for updates to the image a container is instantiated from and automatically pull the update and restart the container using the updated image. However, that doesn't solve the problem of rebuilding your own custom images when there's a change to the upstream image it's based on. You could view this as a two-part problem: (1) knowing when an upstream image has been updated, and (2) doing the actual image rebuild. (1) can be solved fairly easily, but (2) depends a lot on your local build environment/practices, so it's probably much harder to create a generalized solution for that.
If you're able to use Docker Hub's automated builds, the whole problem can be solved relatively cleanly using the repository links feature, which lets you trigger a rebuild automatically when a linked repository (probably an upstream one) is updated. You can also configure a webhook to notify you when an automated build occurs. If you want an email or SMS notification, you could connect the webhook to IFTTT Maker. I found the IFTTT user interface to be kind of confusing, but you would configure the Docker webhook to post to https://maker.ifttt.com/trigger/`docker_xyz_image_built`/with/key/`your_key`.
If you need to build locally, you can at least solve the problem of getting notifications when an upstream image is updated by creating a dummy repo in Docker Hub linked to your repo(s) of interest. The sole purpose of the dummy repo would be to trigger a webhook when it gets rebuilt (which implies one of its linked repos was updated). If you're able to receive this webhook, you could even use that to trigger a rebuild on your side.