For security reasons, php-fpm and nginx run under different users so Nginx-Cache and Nginx-Helper plugins cannot delete the cache on the filesystem.
This is not desirable anyway because it creates an attack opportunity on the filesystems form Wordpress.
A better approach is to use a location-based approach, something like:
location ~ /purge(/.*) {
allow 127.0.0.1;
deny all;
fastcgi_cache_purge tmpcache $1$is_args$args;
}
nginx is already compile with ngx_cache_purge.
Only Nginx-Helper supports this in theory but last time I tried it wasn't reliable.