Redirect after registration and auto login is not working

I was able to get your code to work with the help of the answers to this question. I think you only need wp_set_current_user and wp_set_auth_cookie for the login functionality. Here is my revision of your innermost nested conditional, the rest is the same:

if (!is_wp_error($user_id)) {
              
  wp_set_current_user($user_id);
  wp_set_auth_cookie($user_id);
            
  $url = home_url() . '/redirect';
              
  wp_redirect( $url );
  exit;
}