Grab all of the term ancestors (child and walk up the tree) + the requested metadata.

term-metadata.php

<?php

/**
 * Get the specified metadata value for the term or from
 * one of it's parent terms.
 *
 * @since 1.0.0
 *
 * @param WP_Term $term Term object
 * @param string $meta_key The meta key to retrieve.
 *
 * @return mixed|null
 */
function get_hierarchichal_term_metadata( WP_Term $term, $meta_key ) {

	if ( ! is_taxonomy_hierarchical( $term->taxonomy ) ) {
		return;
	}

	if ( ! has_parent_term( $term ) ) {
		return;
	}

	$term_ancestors = get_terms_ancestory_tree( $term->term_id, $meta_key );
	if ( $term_ancestors === false ) {
		return;
	}

	// Loop through the objects until you find one that has a meta value.
	foreach( (array) $term_ancestors as $term_ancestor ) {
		if ( $term_ancestor->meta_value ) {
			return $term_ancestor->meta_value;
		}
	}

	// Whoops, didn't find one with a value for that meta key.
	return;
}

/**
 * Get an array of term ancestors for the given term id, meaning
 * the SQL query starts at the given term id and then walks up
 * the parent tree as it stores the columns.
 *
 * The result is an array of stdClass objects that have the following:
 *      term_id => int
 *      parent_id => int
 *      meta_value => value of that meta key's column
 *
 * @since 1.0.0
 *
 * @param integer $term_id
 * @param string $meta_key The meta key to retrieve.
 *
 * @return array|bool
 */
function get_terms_ancestory_tree( $term_id, $meta_key ) {
	global $wpdb;

	$sql_query =
"SELECT t.term_id, @parent := t.parent AS parent_id, tm.meta_value
FROM (
	SELECT *
	FROM {$wpdb->term_taxonomy} AS tt
	ORDER BY tt.parent DESC
) AS t
JOIN (
	SELECT @parent := %d
) AS tmp
LEFT JOIN {$wpdb->termmeta} AS tm ON tm.term_id = @parent AND tm.meta_key = %s
WHERE t.term_id = @parent;";

	$sql_query = $wpdb->prepare( $sql_query, $term_id, $meta_key );

	$results = $wpdb->get_results( $sql_query );
	if ( ! $results || ! is_array( $results ) ) {
		return false;
	}

	return $results;
}

/**
 * Checks if the term has a parent.
 *
 * @since 1.0.0
 *
 * @param WP_Term $term Term object.
 *
 * @return bool
 */
function has_parent_term( WP_Term $term ) {
	return ( $term->parent > 0 );
}

This approach grabs all of the database information in one database hit for the given term ID and requested meta key. It's more performant than using a recursive solution like this.