Personally speaking I work for a DNO in Essex and when we attend power cuts in areas that now have FTTC we normally find Openreach engineers attending within an hour or so of us arriving to check the cabinets.

I doubt the cabinet would have battery backup. It would be expensive and still would not provide more then an hour or so's back up. I doubt there is enough room in the cabinett for large batteries and the environmental conditions would meen the batteries would be outside of there operating temperature for much of the time and may even exceed the recommended storage temperature

So in your usual manner you throw out the fact that it has been confirmed by BT employees that there is backup power in the FTTC cabs.
It is most likely to be limited capacity to get round spikes, brownouts etc since the batteries act as big capacitors. VM have installed batteries in their cabs for years. I have kit outside with batteries in far more environmentally hostile situations than the FTTC cabs and provided they are maintained properly function without a hitch.

Even TPON cabinets had battery backup

There are batteries in the cab's 4 in the mini and 6 in the normal DSLAM I think they last about 12 hours but dont quote me on this, I cant remember the exact time but they are mentr for a short period of time

I commission these cabs so I can confirm they do have batteries