Question: In the aftermath of the recent storms in the UK, one of my neighbours reported a flash and loud bang from his local transformer station, and the power briefly cut out. 

After that, several households reported their appliances malfunctioning, including a hair-dryer getting dangerously hot, the smell of burning from sockets and boilers and Xboxes no longer working. The power company cut off power while they investigated and are now repairing/replacing everyone’s appliances at their expense. 

So what would have caused this to happen? I can only imagine a spike in voltage as I understand that breakers and fuses blow on too much current, but I can’t believe there aren’t some safety mechanisms to prevent this from happening.

Answer: 
A broken neutral can cause serious over-voltages on an electricity supply.

UK supplies are usually three-phase and neutral, with approximately 415V between phases, and 240V between any one phase and neutral. The secondary of the substation is "star" wound, with the neutral being the common point. Most UK houses are supplied from one of the phases, plus the neutral.

If the neutral becomes disconnected at the transformer, then there is nothing to keep the phase-neutral voltage at 240V. Instead, it can drift wildly, depending on what appliances are plugged in in each house. The worst case is that one house may see the phase-neutral voltage collapsing to near 0V, while another sees anything up to 415V.

The houses are effectively connected in series between the phases, so the voltage varies with the resistances of whichever appliances are plugged in at the time.
Sounds like either a step down transformer failed and you all got a seriously high voltage and current spike, or it was a lightning strike on the power station.

In either case the surge would likely have shorted out insulation on things like motors. The voltage might well have jumped into the tens of kV briefly. However unlike "normal" surges this would have been accompanied by very high currents.

So, all the breakers fuses etc might well blow, but too late to save some bits of connected equipment.

You could try buying surge protectors, but domestic ones are only rated for quite low energy events. Sticking 10kV at a few hundred amps across it for more than a few milliseconds would have destroyed that as well.