Why Is This Tyre Square On The BA A380?
According to this document about the landing gear system, the wheel dimension  of the main lading gears is 1400x530 R23. This is not the standard notation known from car tires, but it says that the diameter is 1400mm and the width is 530mm, so the tread has a surface of about 2,3m².

The same document further states:


  Depending on the flight phase, a warning will be triggered if a 
  tyre pressure is detected below a given percentage of the nominal tyre pressure (for 5 seconds continuously).
  Likewise, a warning will be triggered if a differential pressure above a given percentage is detected across tyres on the same axle (measured from the tyre with the higher pressure). 


though no concrete threshold is given.

The nominal pressure of the tires is 15Bar (atm) and will quickly fall, even when there's just a tiny leak. So no wonder there was a warning during climb.

During the 13h flight at up to 40.000ft, the pressure could easily fall to the environmental pressure of about 0.2Bar at that altitude. And from that altitude, it takes only 30min to landing (taken from flightaware for a BA32 flight), which may not be enough to equalize tire pressure to ambient pressure through a tiny leak. Also, ambient pressure will rise slowly in the beginning and reach 1Bar at ground quite late.

Considering the pressure in the tire was still 0.2Bar when landing, the pressure difference to the ambient caused a force of 176kN on the tread, which equals a weight of 18t or 39700lbs! (one could also say: 4.5t or 5000lbs on each of the sides of this square shaped tire.)

So it's no wonder that the tire was compressed, though this doesn't explain why the tire choose the square shape.

In general, the tire tries to minimize its volume while the surface is constant, but it can't change to any shape. Its stiffness and the rim only allow the tread to bulge inwards. From math, we know that the more sides a symmetric polygon has, the more surface it has w.r.t its circumference. So the less sides, the better.  A triangular shape seems not to be possible due to the rim and the needed large force to bend side walls. A pentagon doesn't provide enough reduction of volume, so a square is the best choice.

So though the picture looks really crazy, the square shape is not that surprising.

However, a tire does not think about how to collapse best to minimize volume. It starts an one or more points and evolves. If it starts with three bulges, it could have stayed with it, and the A380 had landed with a triangular tire. May be, a fourth bulge has developed at some point. Maybe, the shape in air was different, and some punched during landing pushed the tire into that shape. 

Finally, one can only say that it's clear that the tire collapses to something like this due to the ambient pressure working against the low pressure inside, but one can not really predict the resulting shape. But the rectangular shape seems to be the best to minimize volume.