WordPress 2.7.1 is out..While using upgrade button, it will be stuck at downloading files and will not proceed further. This issue will be faced by people using "Automatic Upgrade" plugin available in prior release. You need to disable the plugin and then re-try the operation. Also note that you have to take a "Backup " of database as the plugin does not take automatic backup.
WordPress Automatic Upgrade plugin did not work for me , as this gave message for my 2.7 installation
"Congratulations!!! Your WordPress version is already up to date"
So you can easily go ahead and disable this plugin and use the Upgrade option available with WordPress 2.7