In the "Form Components" section of the webform, I had a textfield called "Subject".
Because of this no email was sent, and no error was generated. From the logs, it looks like the mail module was not even called.
I spent quite a few hours on this, looking for issues with spam, from email address, etc.
Emails started working after a renamed the textfield "Subject" to "Subject of message".
I did not look into the code, this bug report is purely empirical.