hi,
For account activation email: Please goto wp-admin -> lms -> settings -> emails -> email options -> click on edit activation email button there and check if there is an activation email present or not .
If not then please goto wp-admin -> tools -> buddypress -> reinstall buddypress emails .
Also check if the wordpress wp-login page forget password emails are sending out from system .
Please follow this tip and let me know if still the activation email not working. Tip:https://wplms.io/support/knowledge-base/emails-not-working/
The activation email is sent by the buddypress plugin.
The exact reason why your emails are not sending can be explained by your webhost only.
If you have your cpanel credentials then you can go there and go to WHM and see which email is not sent and why it is failed.
The request is in the pending, this means the buddypress tried to send the email but for some reason it is not sent and this you can see in the WHM.