Handle caching WordPress user counts to avoid slow blocking queries on sites with large user bases. Can be installed directly as an mu-plugin.

tc33-cache-user-count.php

<?php

/**
 * Returns cached user counts if there are some, otherwise fetches current user counts and stores them for later use.
 *
 * @param int       $count Count to be overridden, will be null when called by pre_count_users.
 * @param string    $strategy The computational strategy to use when counting the users.
 * @param int|null  $site_id The site ID to count users for.
 *
 * @return array Includes a grand total and an array of counts indexed by role strings.
 */
function tc33_cached_user_count( $count, $strategy, $site_id ) {
	// Respect any value already set by another filter
	if ( ! is_null( $count ) ) {
		return $count;
	}
	
	$count = get_transient( 'tc33_user_count' );

	if ( $count === false ) {
		// No cached value, so fetch current user count
		$count = tc33_latest_user_count( $strategy, $site_id );
	}

	return $count;
}
add_filter( 'pre_count_users', 'tc33_cached_user_count', 10, 3 );

/**
 * Counts current users as per count_users() and stores the value for use by tc33_cached_user_count() filter.
 *
 * @param string   $strategy Optional. The computational strategy to use when counting the users.
 *                           Accepts either 'time' or 'memory'. Default 'time'.
 * @param int|null $site_id  Optional. The site ID to count users for. Defaults to the current site.
 * @return array Includes a grand total and an array of counts indexed by role strings.
 *
 * @see count_users()
 */
function tc33_latest_user_count( $strategy = 'time', $site_id = null ) {
	// Unhook our filters before fetching the counts
	remove_filter( 'pre_count_users', 'tc33_cached_user_count' );
	$count = count_users( $strategy, $site_id );
	add_filter( 'pre_count_users', 'tc33_cached_user_count', 10, 3 );

	// Save the value in our cache for 12 hours
	set_transient( 'tc33_user_count', $count, 12 * HOUR_IN_SECONDS );

	return $count;
}

Useless for WP6.0+. https://make.wordpress.org/core/2022/05/02/performance-increase-for-sites-with-large-user-counts-now-also-available-on-single-site/

@mircobabini Yes, WordPress have now addressed the issue in core, so this is no longer required.