The White box is an ONT Openreach Optical Network Terminator (ONT). This means that your mum's new flat has FTTP (Fibre to the Premises).
These light indicate power connector, bit like how your router light up or blink. Not all ONTs have battery backup with their unit. These are to ensure in the event of a power failure that your connection doesn't go down. If it says charging, it is most likely charging the backup battery. If it doesn't light up then most likely it isn't charging it or the battery isn't present.
The next photo is the fibre optic external enclosure where the fibre cables enter into your property. And the ports above it are your TV/Radio aerial system, but that doesn't have anything to do with Openreach.