The name suggests that the war took only an hour to begin and end, presumably due to the widespread destruction caused by the massive amounts of weaponry deployed. However, it is possible that there was a cold war leading up to this event and that the "sixty minutes" only refers to the actual fighting.
Following the war came a period of immense geological instability, as earthquakes, tsunamis, glaciers and volcanos swept across Earth. It is uncertain whether this was an unfortunate coincidence, or directly caused by the Sixty Minute War (since weapons technology was greatly advanced at the time of the war, this is not out of the question). The isthmus linking North and South America was destroyed by Slow Bombs, volcanoes sprouted up in Europe and new oceans and mountain ranges were formed.
North America was reduced to a radioactive wasteland, along with much of eastern China. The surviving Chinese fled into the Himalayas and formed a new civilization that would later become Shan Guo, leading nation of the Anti-Traction League. Africa escaped much of the devastation and parts of it became the center of another new civilisation though it was weakened by the Traction Cities centuries later.
This destruction led to the so called "Black Centuries", an era presumably similar to the Dark Ages. During this period, barbaric nomad tribes wandered Europe.
After these centuries (at least 10,000 years after them), Nicholas Quirke transformed London into the first of the Traction Cities and this led to the rise of Municipal Darwinism and the Traction Era.
Several items featured in the series are deadly relics of the conflict, including a directed-energy weapon called MEDUSA which is later mounted upon the Traction City of London. This weapon (and the computer that controls it) was recovered by a group of explorers from an ancient military base on the American continent. MEDUSA is a key element of the plot in Mortal Engines.
Another featured weapon was the device known as ODIN (acronym for Orbital Defence INitiative), an ancient weapon consisting of an particle cannon mounted on a space platform in Earth's orbit. ODIN is a key element of the plot in Infernal Devices and A Darkling Plain, as several major organisations (particularly the Green Storm) attempt to gain control of the device to destroy their enemies.
Other Ancient orbital weapons mentioned in passing are the Diamond Bat, Jinju 14, and the Nine Sisters , although these take no part in the plot and are not described beyond their name. It is implied that they are likely to have fallen out of orbit during the long millennia between the Sixty Minute War and the rise of the Traction Cities.