Change permalink of post if it belongs to custom taxonomy

You can modify the permalinks using the the_permalink filter and add a rewrite rule using add_rewrite_rule to redirect the new permalinks to the standard post link.

function mysite_modify_permalink($url) {
  global $post;
  //if taxonomy archive for Series get term
  $term = is_tax('Series')?get_query_var('term'):'';
  if(empty($term)) { //if no term was found
    //get all Series terms for current post
    $terms = get_the_terms($post, 'Series');
    //if terms were found, take first term found
    if(!empty($terms) && is_array($terms)) $term = $terms[0]->slug;
  }
  //if term was found return a modified permalink url
  //(replace last url segment starting from "https://wordpress.stackexchange.com/" with '/<term slug><last segment>')
  if(!empty($term)) return preg_replace('#/blog(/[^/]+/?)$#', '/blog/'.$term.'$1', $url);
  //else return original url
  else return $url;
}
add_filter('the_permalink', 'mysite_modify_permalink');

function mysite_add_rewrule() {
  //get all Series terms
  $terms = get_terms(array('taxonomy' => 'Series'));
  //if terms were found, add a rewrite rule for urls containing one of them
  if(!empty($terms) && is_array($terms)) {
    foreach($terms as $key => $term) $terms[$key] = $term->slug;
    $terms="(?:" . implode('|', $terms) . ')';
    add_rewrite_rule('^blog/'.$terms.'(/[^/]+/?)$', 'index.php?name=$matches[1]', 'top');
  }
}
add_action('init', 'mysite_add_rewrule');