I have Caller Display, and it's working perfectly.
However, having read this thread, out of curiosity I tried dialling *#001#, and all I get is the message saying "Sorry, this service requested is not available."
I would suggest there's a fault on the *#001# system.
Other than seeing it in a previous post I have never heard of *#001# as a BT service code. And I worked in Telecomms for years.