It's a unfair to say that the countries of East Asia didn't have an industrial revolution because they did, just later than England.
One of the main reasons England was able to industrialize first was because they had an obscene amount of capital that they amassed from exploiting their colonies (especially India). They also were able to import food from the colonies, which made it feasible for people to come off of the land and to go work in the factories. They also had steam power to exploit to power their factories.
Korea wasn't able to industrialize because it didn't have the capital. Japan, thanks to being a feudal state already had the systems in place in order to modernize/industrialize during the Meiji period. Because Feudal lords were always in competition with each other, Japan had already built up a large amount of capital necessary to being industrialization.
Korea and China, on the other hand, didn't have that. They both (especially China) had large, centralized governments. People depended on the land to survive, and wouldn't have been easily convinced to give it up to build factories. China had, for thousands of years, been the head of the region and had many countries paying tribute to it. However, in both Korea and China there was a lot of corruption within the city officials, so a lot of money was siphoned off.
China and Korea did not value merchants in their confucian system. They were at the bottom of the hierarchy, with scholars at the top. Making money was considered to be unconfucian, and there wasn't a large incentive on making capital because of this.
The tl;dr here is that England had systems in place through their empire that allowed it to industrialize first. The East Asian countries did so later, they didn't simply not have an industrial revolution.
This is the question that has been debated by historians over and over again, and there are several arguments that have been made, which I will highlight. What you are referring to has been called 'The Great Divergence.'
Geoffrey Parker, in his recent book Global Crisis: War, Climate Change, and Catastrophe in the Seventeenth Century argues that Europe developed superior educational systems that essentially allowed for what we now call 'academic freedom.' This meant that scientific discovery was more frequent, and less prone to what he calls "accidents." While I personally disagree with this argument for China's lack of industrial revolution during the nineteenth century, it does make sense for Japan.
Japan's lack of industrial revolution most likely stems from economic and intellectual isolation from just about everyone except the Dutch. It was not until Commodore Perry opened Japan that they were more or less forced into accepting the West. This has been argued by Kenneth Pyle in The Making of Modern Japan
Another argument was made by Mark Elvin in The Pattern of the Chinese Past where he states that China did not go through industrialization because it had already gone through what he calls an 'economic revolution.' The Chinese already had markets, and advanced methods of transporting goods across their sovereignty, and the continent. This means the Chinese felt no need to industrialize, since economically they were doing quite well (when you exclude the seventeenth century, which was a period of global famine, disease, death, wars, etc.)
I personally cannot speak for Korea, but hopefully someone else has the answer.
If you are interested in this question, I would read the following sources:
Well, since a couple people said they couldn't answer for Korea, I'll take a crack at it.
The two major powers east Asia had closed their borders to foreign interference, and Korea, stuck between these two countries, and in tributary status to one of them (China), followed their lead.
China and Japan gained their industrial revolutions because outside powers used military force to open up trading between them. The First and Second Opium war between England and China led to China's defeat, and forced them to sign unequal treaties leading them to open several of their cities up for foreigners, and included a one hundred year lease of Hong Kong. Commodore Matthew Perry opened Japan up for trade, by making them sign unequal treaties as well.
Some of the ruling class in Korea saw the writing on the wall, and wanted Korea to modernize as well, which led to the Gwangmu reforms, but were quickly shot down by the royal family, and had most of the reformers branded as traitors. Japan, on the other hand, saw the advantages of becoming an Imperial power, and the disadvantages of becoming a colony, and sent out the Iwakura misison to teach them how to modernize and become an Imperial power as well. Shortly after this, Japan won the Sino-Japanese war, and then the Russo-Japanese war, eliminating any power who would attempt to claim Korea as their own. Japan made Korea a protectorate in 1905, and then annexed it in 1910. Japan wanted to develop Korea as a colony, and spent a great deal of resources modernizing Korea with railways, cars, electricity, mass media, and universal education.
TL;DR China and Japan were modernized by being forced to trade with foriegn powers. Korea was modernized by Japan.