OK. Found the problem.
When sites are moved between web hosts, or generally in the case of WordPress not being able to create a new monthly uploads directory, the internal WP core function wp_upload_dir will return only an error, and not any useful information about existing uploads. This prevents AutoNav from discovering where your uploads directory is at all, which is why things break.
WP 3.5 and later properly return both "where uploads are supposed to go" in addition to an error message.
Version 1.5.3 of AutoNav will show WP 3.5 as the minimum version, and will also be more graceful in simply not displaying images for the case of WordPress 3.4.2 and below.
- Upgrade to the latest WordPress
- Correct the permissions on your uploads directory.