Solved it!
For some reason, PHP’s mail() function was hanging. I changed line 189 of wp-includes/pluggable.php from:
$phpmailer->IsMail(); // set mailer to use php mail()
to
$phpmailer->IsSendmail(); // set mailer to use sendmail
Of course, I made sure I had a working sendmail first. All good, thanks to everyone that pointed me towards email as the culprit…