Customers not notified.
THE FIRST CHECK
cPanel > Track Delivery. Search for a recent customer address.
If nothing appears, the mail never left the server, which points at the application.
If it shows success, the message was accepted and the problem is at the recipient's end.
THE MOST COMMON CAUSE
The application uses PHP mail by default. Configure SMTP with a mailbox on your own domain instead.
THE FROM ADDRESS
Must be on your domain. Sending as the customer's address fails authentication.
AUTHENTICATION
cPanel > Email Deliverability. Repair SPF and DKIM.
WOOCOMMERCE SPECIFICALLY
Check WooCommerce > Settings > Emails. Confirm the relevant notifications are enabled and the recipient addresses are correct.
Administrator notifications go to the address configured there, which is frequently wrong after a handover.
THE CRON DEPENDENCY
Some email is queued and sent by scheduled task. Without a real cron job, the queue never processes.
TESTING
Place a test order and confirm both the customer and admin notifications arrive.