Maybe the cache tables are out of date, particularly cache_workflow_ng. I've seen that cause some very strange things. I hope that's it because I don't see what else can be wrong. Checking mod_ssl was going to be my next suggestion.

At this point it might be time to put some debug lines in the code itself. I'd start at the beginning of uc_ups_quote() to see if it's actually being called ever. Use drupal_set_message() so it won't mess up the normal functioning, though you'll have to refresh the cart page to see the messages.