Step 1: High Cost of Labor and Money
The Greeks had invented a primitive steam engine over two thousand years ago, but absolutely nothing came of it. Why? Because they had slaves. Slaves are really cheap labor and they can get enough work done that you can sit back and enjoy philosophy and pederasty all day without ever having to actually create new solutions to get work done. This continued well past the Fall of (Western) Rome, in which feudal lords had serfs, who were again cheap labor that lent plenty of time to enjoy theology and jousting. But this changed as feudalism fell apart in Western Europe, and soon you had a much more monetary economy. With mercantilism and eventually capitalism workers were actually being paid in money, and there was a greater demand for materials (I’ll get to that next) and labor was again more expensive so labor saving inventions were heavily sought after and got financial backing. For example, the steam engine was invented to pump water out of coal mines because coal was a valuable commodity and it cost money to get workers to otherwise remove the water. So your Dwarves could get this part done earlier by having labor be more expensive. This could be accomplished via an earlier monetary economy that lacks large scale coerced labor which isn’t terribly hard to conceive of given the existence of merchant republics in the Middle Ages.
Step 1.5: Copy Someone Else/Haul Ass
In 1860 Japan was a literal feudal nation, with samurai and all up until Commodore Perry showed up and demanded that Japan play ball with America. But within a hundred years they were bombing American ships. This is because Emperor Meji did not fuck around. The Emperor sent out Agents all over the world to learn about industrial sciences and they caught up extremely quickly. Stalin and Mao likewise overhauled their countries to become extremely industrialized through forcible collectivization and labor campaigns. Granted, these things are not so much an industrial revolution (ie invention of such technology) but the rapid industrialization of the country (ie the implementation of these technologies)
Step 2: Enough Food.
The Industrial Revolution was preceded by the Second Agricultural Revolution. The British developed or adapted several breakthrough techniques and technology that allowed agricultural production to skyrocket to levels never seen before, with everything from advanced crop rotation to seed drills, to the repealing of the corn laws, to potatoes. Speaking of potatoes, they lead to a veritable population boom in Europe. If you are able to produce an excess of food, that means more people can move to the city to seek other jobs, and this gives more opportunities for people to either invent something that saves labor or they can work 13 hours a day on it once it’s made. So have a Dwarven equivalent of a potato.
Step 3: Motivation
The Spinning Jenny wasn’t invented for fun, it was invented to spin cotton so you could sell more of it faster and cheaper. Likewise industrialization efforts in Japan and Russia were done with the explicit goal of catching up to nations that had already industrialized. So you’re dwarves need to be motivated to innovate, which you’ve provided.
So to recap, your dwarves need an economy that doesn’t have a bunch of slaves and serfs, has a powerful ruler that gets shit done, has enough food not to starve, and motive.
How I could see it play out: The Dwarves are a reclusive kingdom that is very centralized compared to its feudal neighbors, with an absolute monarch more evocative of an early modern period king, or perhaps a Roman Emperor. The Dwarves primarily export iron and coal from the mines in the rugged mountains, and they need to burn a lot of coal and wood. To save time and effort they use the fast flowing rivers to power sawmills and stampmills to cut lumber and pound iron respectively. The dwarves are master blacksmiths, but hammering is exhausting, so water hammers make life easier, which was a niche and expensive tool before the Great War, but soon they needed to produce hundreds of blades monthly, so they built more, and they also needed more coal, but their good coal mines were exhausted, and now they had to get it from these awful ones prone to flooding...