Did the UKBA give you a tracking number (usually to the UnitedParcelService)? That would be for overnight and I can't imagine why they wouldn't use it because applicants pay for expedited return when the application is paid for online.
And I am so-so-so-so-so sorry to hear about your daughter's visa. As others have posted, don't give up quite yet. I can't believe you would only have a four day appeal. More information should be in the letter from the UKBA discussing the determination.