On 17th of July, Robert Haas committed patch: Add new function pg_notification_queue_usage. This tells you what fraction of NOTIFY's queue is currently filled. Brendan Jurd, reviewed by Merlin Moncure and Gurjeet Singh. A few further tweaks by me.
Similar question has been asked many times on mailing lists and on IRC. Sometimes it's not mail sending, but file/directory creation, or something else that generally requires some interaction with "world outside of database". Can it be done? Sure. How, then?