WordPress won't automatically update

If for some reason auto update does work on your server, have you considered updating WP manually?

Download a fresh copy of the WordPress .zip file to your computer, unzip it, and use that to copy up all files and folders EXCEPT the wp-config.php file and the /wp-content/ directory. You may need to delete the old wp-admin and wp-includes folders and files on your server before uploading the new ones. Please read the Manual Update directions first.
– Backup: If you haven’t already done, always backup everything (including your database) before doing any actions, just in case something really goes wrong. You can never have enough backups!
Useful Guides: http://codex.ww.wp.xz.cn/WordPress_Backups ; http://codex.ww.wp.xz.cn/Backing_Up_Your_Database ; http://codex.ww.wp.xz.cn/Restoring_Your_Database_From_Backup
– Memory: try increasing memory available for php.