Snippet to send estimated delivery on top to admin for new order and to customer for processing order status. Estimated Delivery Date Plugin for WooCommerce by PluginHive - https://www.pluginhive.com/product/estimated-delivery-date-plugin-woocommerce/

functions.php

/**
 * Snippet to send estimated delivery on top to admin for new order and to customer for processing order status.
 * Created at : 30 July 2018
 * Updated at : 30 July 2018
 * PluginHive Plugins : https://www.pluginhive.com/plugins/
 * Gist Link : https://gist.github.com/xadapter/9a20d5f173498dd8a68074d4b87dafb1
 */

add_action( 'woocommerce_email_order_details', 'ph_add_estimated_delivery_after_heading', 1, 4 );

if( ! function_exists('ph_add_estimated_delivery_after_heading') ) {
	function ph_add_estimated_delivery_after_heading( $order, $sent_to_admin, $plain_text, $email ) {

		// Supported Tags [DATE] in case of simple, [DATE_1] and [DATE_2] in case of date range and simple range
		$estimated_delivery_text 	= "<strong>Estimated delivery :</strong> [DATE_1] - [DATE_2]<br/><br/>";	// Text to be printed on email html supported
		$estimated_delivery_format 	= 'date_range';	// Possible value - simple or simple_range or date_range

		// Attach estimated delivery to email sent to admin for new order and Customer for processing order status
		if(  $email instanceof WC_Email_New_Order || $email instanceof WC_Email_Customer_Processing_Order ) {
			$estimated_delivery = $order->get_meta("_est_date");		// Get estimated delivery from order

			if( ! empty($estimated_delivery) )
			{
				// Simple range and date range
				if( $estimated_delivery_format == 'date_range' || $estimated_delivery_format == 'simple_range' ) {
					$exploded_est_delivery = array_map( 'trim', explode( '-', $estimated_delivery ) );
					// Two date or days count should be available in case of range
					if( count($exploded_est_delivery) == 2 ) {
						// Remove days in case of simple range
						if( $estimated_delivery_format == 'simple_range' ) {
							$exploded_est_delivery[1] = str_replace( " days", "", $exploded_est_delivery[1] );
						}
						$message = str_replace( array( "[DATE_1]", "[DATE_2]"), array( $exploded_est_delivery[0], $exploded_est_delivery[1] ), $estimated_delivery_text );
					}
				}
				// Simple
				elseif( $estimated_delivery_format == "simple" ) {
					$estimated_delivery = trim($estimated_delivery);
					$message = str_replace( "[DATE]", $estimated_delivery, $estimated_delivery_text );
				}
				// Print the message in email sent
				if( ! empty($message) ) {
					echo $message;
				}
			}
		}
		
	}
}