Actually, the requisite parts of industrial revolution technology were present but no Aristotle-like genius put them together. The Romans and the Helenistic technicians of Alexandria knew a lot about hydrolics, and Hero of Alexandria had demonstrated that mechanical force could be generated by heating water in a closed vessel. In the Roman city of Bath, there is a first or second century double cylinder water pump made of bronze that could easily have been scaled up to be the guts of a steam-powered engine. (Remember that Watt's original steam engine in the late 18th century was built to pump water out of coal mines.)
How about this for a time line?
150 AD - Technicians in Alexandria combine Hero's steam-powered toy to provide power to operate the above-mentioned twin cylinder water pump in order to provide water to the upper stories of wealthy houses and to move water uphill to remote areas where aquaducts were impractical.
190 AD - Steam engine is combined with the Archimedian screw (another water-raising device)
200 AD Someone realizes that a powered screw that raises water against gravity could push a ship through the water. over the next 50 years, steam powered vessels become common
Around the same time, recognition of the ability to operate industrial scale grain milling and metal forging is discovered, decreasing the cost of food and avoiding the long delays in moving food and manufactured goods around the Mediteranian world. The improvement in the economy combined with the gradual emergence of a technical/middle class staves off the economic crises of the third and fourth centuries. When the barbarian tribes attempt to invade the Empire, they find a richer and less stratified Roman world than IOTL.
By 1100 voyages across the Atlantic and Indian Oceans have extended Roman culture to the Americas and India and beyond.