How to send Woocommerce Customer Details (Name, Email and Phone Number) to Admin upon signup

Try changing this: $name = $user->user_email; to this:

$name = $user->first_name;

And this: $phone = get_user_meta( $customer_id, ‘billing_phone’, true ); to this:

$phone = get_user_meta( $user_id, 'billing_phone', true );

Did it work for you? Let me know! =)