Turns out it was a simple matter of switching my server to php5, even though I thought I was already using php5. Oh well, live and learn I suppose. And maybe take a lesson from President Reagan, “trust but verify”.
How long does it take to get a “Success” message, or an error message? I only get to the “Downloading update from http://wordpress.org/wordpress-2.7.1.zip” line and then nothing happens. No progress meters, status updates, nothing. I never used the auto update plugin that was available previously, so that is not my issue.
