The short answer: mastic asphalt is bitumen mixed with fine stone, heated until it flows, and spread by hand into a dense jointless mass that waterproofs by being solid rather than by being sealed. London is full of it because it was the material of choice while the city was being rebuilt and extended, and a great deal of what was laid then is still up there.
What is the material, exactly?
A blend rather than a sheet. Bitumen supplies the waterproofing and the flexibility, the stone supplies body and strength, and the proportions are chosen so the mix flows when hot and stays put when cold. It arrives on site in blocks, goes into a mixer that keeps it moving at temperature, and is carried across the roof and spread with a wooden float.
That last part is why it behaves so differently from everything laid since. Nothing on the roof is a joint, so nothing on the roof is a joint that can open: what covers the middle also climbs the walls and wraps every obstruction, all of it one body of material. Laying it is not assembly. It is floating a single surface into place.
Why did London end up with so much of it?
Because it suited the moment. Asphalt was already the trusted flat roofing material when the city needed an enormous amount of flat roofing quickly and expected it to last: the post-war rebuild, the estates that followed, the school and hospital wings, the parades of shops with flats above, and the countless rear additions bolted onto Victorian terraces in every borough.
It also suited the buildings themselves. Concrete decks were common, and asphalt sits happily on concrete. Parapets were common, and asphalt turns up a parapet without a detail to design. And it was laid by a trade that existed in numbers at the time. The result is that a surprising share of the flat roofs in South London are still the original asphalt, quietly doing the job somebody expected of them decades ago.
How does it fail?
By stiffening rather than by splitting open. Bitumen gives up its flexibility after enough decades of heat, frost and ultraviolet light, and a hardened surface has nothing left with which to follow the building as it shifts a fraction each day. So the failures appear where movement concentrates: at a change of plane, at a junction with brickwork, at an edge. The middle of the field is usually the last part to give trouble, which is the reverse of most people's expectation.
Surface crazing is the earlier warning, and it means something different: it says the protective finish has weathered away and sunlight now falls directly on the bitumen. Shallow at first, and worth acting on before it stops being shallow.
What are the honest options now?
Better ones than the material's age suggests. Asphalt is unusual in supporting genuine repair, because the mend is made from the material it is mending: the defect is opened out, brought back up to working heat and reworked until the new asphalt and the old are one body again. Where the rest of the roof is sound, nothing beyond that is called for.
Where the stiffening has reached the whole roof instead of one junction, the choice between renewing the surface in place and taking the roof off is a real decision with evidence behind it, and it is set out on the mastic asphalt page rather than repeated here. What decides it is not the age of the asphalt but what a survey finds underneath it.