this documents whether or not you've received the Mail Tutorial

Messages about the game will appear in this tab. Here on the 'Recently' panel, you will see information about things that have happened – new developments in stories you have played, consequences that have come about, or interactions with other players. On the Messages panel, you will see invitations from other players that require your attention. Your 'A Message Awaits' Quality has gone!

don't remember what activates this or how long it takes; both "Contemplate your incredible good fortune" and "Hit the cobbles" trigger living stories with Envelope.png icons.

I am however leaning towards "Hit the cobbles", as the older, retired equivalent of "Contemplate your incredible good fortune" ("Explore Fallen London") activated the Mysterious Benefactor living story (which also uses the envelope icon)