How to run custom command running drush cron via sudo?
OK, I've finally solved my own problem.

In summary, it wasn't actually a Drush thing in the end.

In detail, my custom drush command was also fixing file permissions and ownership, as per https://drupal.org/node/244924 (because cron can sometimes create a file, and then the file may have the wrong owner/perms).  This sets the ctime on all files, and it turns out that Drupal update checking relies on the ctime.

With hindsight of course I should have done a test with the permissions line removed from my custom command, but it hadn't occurred to me it could be relevant.