Updating programmatically customer's billing information in WooCommerce
You can do it using update_user_meta() function, this way:
$user_id = get_current_user_id();$data = array( 'billing_city' => $city_value, 'billing_postcode' => $postcode_value, 'billing_email' => $email_value, 'billing_phone' => $phone_value,);foreach ($data as $meta_key => $meta_value ) { update_user_meta( $user_id, $meta_key, $meta_value );}You will need to set the values in the array.
You have to save the changes after setting properties. In your code, after the foreach, add:
$customer->save();And voila!