WordPress- WP Not Sending Activation Email is a problem which more developers don’t know how to fix it.
If your website sending email system is working fine, but the Activation or set password email for a new member not sent, and you don’t want to use SMTP service.
You just need to make a little change into a wordpress php file named (( pluggable.php )).
The file “pluggable.php” is located into the folder (( wp-includes )). Follow these steps to fix this Issue:
wp_mail($user->user_email, sprintf(__('[%s] Your username and password info'), $blogname), $message);
Also, you can use a plugin which allows you to send emails into your wordpress using SMTP service, like ( Easy WP SMTP plugin ).