In reply to J.Duncan - Paperless Creations.

There is a plugin I installed on a former client that is supposed to e-mail the admin address whenever there is an update. I can't remember if WP Core, themes or plugins.

Since that client is a former client (let me go a week after the plugin was installed). I have no idea how that plugin works.

When I get up to my laptop every morning, I log in on all the sites I manage (client and my own) and check for updates.