Update Drupal Core from 6.26 to 6.36 using update.php gave no errors, but Drupal still reports that it is on version 6.26
Have you manually brought your codebase up to version 6.36 by overlaying 6.24 with 6.36?

The update.php will only update the database to the latest version of Drupal installed.  So if it's still 6.24, it makes sense to me that it is applying 0 of 0.

https://www.drupal.org/node/1223018