I did what any right minded person would do - took it apart to see how it works.
OK, so modern electronic gadgets aren't easy to disassemble and reassemble. Thankfully I had no great desire to put it all back together again. One nice thing was that the UV photodiode was on its own board. One interesting thing is that there's a skeleton potentiometer so presumably these things are calibrated/adjusted at the factory.
Google led me to this manufacturer as the likely source of the photodiode http://www.eoc-inc.com/UV-detector-GaN.htm and here's the datasheet for a device designed for UV Index monitoring http://www.eoc-inc.com/genicom/GUVB-S11SD.pdf
Good news is that device current increases linearly with UV Index, bad news is it's measured in nano amperes. Well I suppose that does mean the resulting device could be very low current; given the application it might as well be solar powered!
First things first, I'm going to need an amplifier, so I went for a low power device I bought a few of a while ago but lacked the frequency response I needed, the TLV2252. Hooking this up on a breadboard with a circuit based on this one from http://www.ti.com/lit/an/sboa035/sboa035.pdf I used 10M resistors and a 3.3V supply.
Covering and exposing the photodiode did nothing, so I started thinking I'd damaged it or the circuit wasn't right. I swapped the UV photodiode for an LED and it worked like magic throwing the output to max when I directed a bright torch at it, so I swapped back the UV photodiode and went outside. Yes it worked giving about 20mV per UVI. OK, that's not a lot, but there's a second OP amp in the package so I used a 10K and 100K resistor to give extra gain of 10 and hey presto it's ready to interface to a ADC. As it is the circuit draws less than 0.2 mA.