My upgrade hangs on “Downloading update from https://www.ads-software.com/wordpress-2.7.1.zip” as well.
Edit: My upgrade worked. I previously assumed that something would happen beneath the above-mentioned status message. After waiting and nothing happening, I clicked the “WordPress 2.7.1 is available! Please update now.” link at the top of the page and the upgrade process started. I had assumed this link would take me to the same page I was already on.
Perhaps this process can be clarified for future versions. When something says “Downloading update” and nothing changes, it’s natural to assume that you shouldn’t click anything, as it would disrupt the download.