The New Order email is a special email type when compared to others.
The primary difference is that: It is NOT triggered based on the order status. The email is sent only to the recipients specified in the WooCommerce email settings.
Let us look into this in detail.
What happens when an order is placed?
When a customer places an order, WooCommerce sends email notifications to both the store administrator and also to the customer.
For Customers :
The customer would receive an email based on the status of the order.
Example: If he made the payment successfully, WooCommerce marks the order as Completed. This is the normal order status (unless your payment plugin has some special workflow). So the customer gets the Completed Order email template
Go to WooCommerce -> Settings -> Emails -> Completed Order email.
Make sure you have enabled it and added a subject and an email heading.
Also make sure that you have set it as HTML.
For Store administrators:
The Store administrators would normally get the New Order email notification. As mentioned earlier, this is a bit special type.
You need to enter the emails of the recipients in WooCommerce -> Settings -> Emails -> New Order emails.
NOTE: Only when you enter the recipients there, you would get this email when an order is placed. Otherwise you would not get an email
What is the role of WooCommerce Email Customizer ?
The email customizer plugin only takes care of the design of the email body. It supplies the designed email template to WooCommerce.
It is WooCommerce that sends the emails. Yes. You read it correct. WooCommerce handles the email sending part .
Click here to read more on troubleshooting WooCommerce Emails