I have a 2012 A6 Avant S-Line 3.0TDi, lovely car, only 68K miles, I've owned it since 2013. Up to now, the only thing I've ever had to replace was the rear wiper motor.

I use hands free quite a lot, and last week, out of the blue, it stopped working. 

Having established that my phone works OK in wifey's car, I deleted all phone data from the Audi, and started again. But it will not detect my iPhone, nor any other iPhone, nor an Android phone.
In the menus, it looks like Bluetooth thinks it's all working.
So, what next? Sounds like it's the component that sends/receives the Bluetooth signal that has failed. Any ideas where it is please? Or whether it's feasible to replace it or get it replaced?
Afterthought........I suppose it couldn't have its own fuse could it? Everything else appears to work............

As a follow-up to this query -

The car went into our local Audi independant specialst last week for its MOT and I asked them to take a quick look at this problem. They reckoned it's a problem in the module that houses the aerial, which they said is beneath the floor and hard to get at.

Does anyone know where? I took a quick look and found a small compartment beneath the driver's seat (RHD) but it only appears to contain connectors for the seat electrics. All looked OK in it too.

Sorry to hear of your issue. I suspect it's under the carpet unfortunately.