I got a callback from the person who I spoke to at United Healthcare yesterday to let me know what is happening.
Apparently, everyone agrees that my sleep study should be approved, but there is something that needs to be done to actually get the approval into their system that is being problematic. It's not clear what that problem is from my end of the world (and perhaps not from *her* end of the world), but I am told that I should hear back about this being approved today or tomorrow.
At which point, since the approval is set to expire on November 12th (because it would be a three month approval from the original date of the request), we will have one month to get the sleep study done.
We'll see what happens next.