Update: Looks like the workaround doesn't offer a fix for everyone. This will speed up your download of the IPSW to your computer. But you may still run into authentication issues while Apple's servers are being hammered. Looks like our original advice to be patient is still valid.
Good news everyone. If you're experiencing the horror of Error 3200 instead of dancing in the technological flowers of iOS 5, there's a workaround discovered by GDGT users.
It's as simple as downloading the iOS 5 IPSW (iPhoneSoftware) for your device directly from Apple instead of downloading it from within iTunes.
The IPSW is the file that contains the operating system for iPhones, iPads and the iPod touch. After you've downloaded the appropriate file from the list below, here's how to update your device.
Plug in your device and Option+click on the Check for Update button in iTunes. If you want to wipe your device and do a clean install, you can Option+click on Restore.
A window will pop up, navigate to the IPSW you downloaded and select it. That's it, your device should begin to update.
Here are the iOS 5 IPSW files for each device.