now comes the awkward moment when I advise that I work for Nationwide in one of the mortgage centres!
now obviously I can't speak for your case since I've not dealt with it personally, but I do find it weird when they said they can't fax references, I mean in my department we want fax numbers or email addresses so we don't delay things unnecessarily.
I'm assuming from your messages that you went through a branch directly (lack of mention of financial adviser) and recently we've been in the process of moving branches to our new system (not excusing service issues).
I'm back at work tomorrow, so if you want me to have a look at your case and see where it's at you can PM me your reference number (should start either M2 or M55). Of course I'll understand if you don't want some random guy looking up your details!