I have started receiving notifications again! Like others, I did not have any options in notification settings that related to emailing messages.
It looks like Facebook integrated chat with messaging. I discovered that when logging into Facebook, missed messages would appear as chat windows. After turning off Chat (chat window > bottom right > settings cog) through the web, turning off chat and notifications on my phone, and turning off chat on my messenger clients, I'm starting to get emails again of messages.
Not sure if this kicks in immediately, because I got my first notification 2 days after I changed this. Note that the message was sent, and email arrive 30 mins later.
I also submitted a report about this, and you can too if you like:
... I didn't receive notification
... File a report