In my world, there is a zombie pandemic that happens. The zombies are fast, but can be defeated by a large army. My world a little before the outbreak was close to having an Industrial Revolution. They had good agriculture, had a capitalist way of thinking, a system similar to the putting out system, and begun to mess a little with machinery. Is it still possible for them to industrialize, and how?
It may help an advanced civilization to go underground to quarantine against zombie infection. Technology will develop because they will be living and working in modified mines.
Zombies may be fast, but almost certainly have lost some of their mental capacity (due to the lizard brain thing discussed by many zombie experts). They will either lay in wait or wander randomly. Zombies, like other creatures, will need some form of food to survive although they may be more able to survive on reserve capacities or in a near hibernating state when near starvation.
Limitations of your Civilization
Many civilizations develop mining skills quite early, and are capable of living in caves far before then. Even pre-industrialization the ideas of systematic quarantine was understood as a means to control infection. The combination of quarantine with mining, and perhaps a system of surface holes which lets in surface daylight (possibly covered with glass, which was also developed pretty early and was pre-industrialization) may allow underground agriculture.
The individual communities, because of quarantine-style community separation, will be cut off from communication and exchange with other communities. Each mining city will need to be self-sufficient in some way. If goods need to be exchanged they may have to take a risk using the basket system. Just outside the city gates the exchange product is lain, with the request for the product that is needed. The baskets must then be exchanged with no direct contact. This is supposed to reduce the threat of becoming a zombie.
How Technology May Develop
The earliest forms of industrialization began with water power. Underground streams, springs, and rivers can be controlled for these purposes. Metal casting and blacksmithing will eventually give way to metal hammer forges (hydraulic or steam mechanic presses), which will make robust mechanization of tasks possible. Fuel could be charcoal from trees harvested from the surface, or coal from the mines (indicating rich deposits at different layers, so it must be a geologically unique area for industrialization to work well).
Their thinking, although capitalistic, may also have to adapt to resilience and long-term resource sustainability. This is because without vast interconnected highways and waterways it is very possible to deplete resources too rapidly. This indicates that the actual pace of work within these underground cities will be slower than most modern civilizations.
The underground citizens will be very aware of the zombie threat and will be working towards various methods to keep safe and to defend themselves should zombies come within their perimeters. Eventually they may have to join with other underground cities to fight them as an army (once it becomes apparent that they have the technological means to do so). The avenue by which this happens must be nearly accidental, since this will be so different than the way in which they have done things before.
Perhaps something will cause the dynamic with the zombies to shift. This could include some person of malicious intent deciding to use them as a weapon or some kind or evolution in the zombies that cause them to be more of a threat. Whichever is the case, it will be necessary to band together to survive, which will lead to the vast overground war that you are describing.
Heck, the zombies might spur the industrialization, as a threat.
Think of the zombies as a kind of predator. Early farming humans built walled villages and homes, primarily to protect themselves against predators (including other humans). The walls were made of wood, mud bricks, stones, eventually tall enough and thick enough to protect them.
Industrialization is just the beginning of automation of some prevalent activity that is done frequently. The ideas of industry, assembly lines, and machine assisted mass production begins with some sort of engine.
IRL, that was capturing the natural energy of wind and rivers. The word "mill" comes from ancient roots meaning to grind, as a place it was a building to grind grain (circa 900AD) into a powder (aka flour). The first "industrialization" was indeed about using a water wheel turned by a river or stream as the power source to turn heavy stones that quickly ground grain into flour. Later this was adapted to turn saws, and became the "lumber mill". Elsewhere, on coasts where the wind was fairly reliable on most days, "wind mills" were used both for grinding and for pumping water. In essence the nature of the word "mill" came to mean something besides grinding, it was a building designed to capture energy to turn something.
Many people think textiles were the beginning of industrialization; along with steam engines, but in truth the water wheel at least a thousand years before that was a breakthrough in using non-animal power.
Amid a zombie outbreak, the primary concern of humans is to isolate and protect themselves from zombies, and some clever industrialization and automation can help that. Nobody resists the automation because they all want to live.
Building walls, digging deep ditches, routing streams. Building weapons, swords, knives and nets. Building traps. Building alarm systems. Boiling water for drinking and cooking.
Farming, fishing and hunting can be industrialized. The threat of zombies means we suddenly have far more work to do than we have people to do it; and part of the answer is turning some of that work over to machines that we build to do it. This can be the spur for some genius to invent the steam engine.
I'd begin with a small walled settlement straddling a stream, so we can use a water wheel for power, and even rig it to catch fish for food. (It scoops up the big-enough fish in woven baskets that drain out the water, then as the wheel raises them, the turn of the wheel tips the basket of still live fish into a chute leading to a shore-side pool where the fish will be easy to net as needed.)
That should solve nearly all of the daily protein problem, at least.
I'd posit that after that, the industrialization extends to extending the walls of the village, up and down the stream. Perhaps invent steam engines, and build out, away from the river. Steam engines can be used to both dig (wells, mining) and to pump water, and good well-sites are typically plentiful even a few miles from a stream or river. The reason for moving away might be to farm, but any farm needs to fenced off from zombies, and that's a huge amount of labor best done by machines.
In short, protecting ourselves from a zombie threat can be the catalyst for industrialization, creating factories, and multiplying the effectiveness of people by using machines. Not an impediment to industrialization at all.
Look at the USA in WW II. The entire country mobilized to fight Hitler, the men went to war, the women, for the first time, went to work en masse as factory workers producing weapons. There was enormous cognitive dissonance in the country, Rosie the Riveter and all that, females doing that kind of heavy lifting machine factory work was unheard of, and then suddenly common.
Why the sudden upending? The threat of Germany taking over the frikkin' world, conquering everything if we did not. That external threat forced changes to all of American society and the roles of men and women, permanently. And not just in America, in Europe and most of the world.
The Zombie outbreak is a similar existential threat, and a very good spur for people to consider a whole new model of their society, just to survive.
Zombies are a potential source of limitless labour
If your zombies are physics busting creatures that seek brains, they can be harnessed to power devices. They can replace fuel sources like coal and help power machines to run industrialization.
Rather than destroying any chance at industrialization, they could enable it.
You can put them in wheel or cranks, dangle a brain in front of them, and power all sorts of machinery.
Anyone who dies can donate their body to industry for a good fee.
Industrial Revolution will be delayed
When there is threat to survival, the first thing people will do is to save their lives. As told by Amadeus, in WW II, the entire country of USA started to produce weapons. As told by the U.S. Department of Defense
Prior to World War II, factories in the United States were turning out automobiles, large and small appliances, and childrens' toys.
In January 1942 — a mere month after the attack on Pearl Harbor, Hawaii — President Franklin D. Roosevelt ordered the establishment of the War Production Board.
Its purpose was to convert the factories of peacetime industries into manufacturing plants for weapons and military equipment for the fight.
The same thing will happen in your world. All the factories will start making weapons and forget other industrial things. So even if your world "was close to having an Industrial Revolution", they will convert all industry into manufacturing plants for weapons.
The Industrial Revolution will be delayed until the danger is over.
Or your equivalent island full of tea drinking scientists and engineers. Because they manage to purge the zombies from their island with the St Patrick maneuver borrowed from the Irish (they are ok too) snake ridding technique, followed by careful screening of incoming ships (each incoming crew member in turn dances naked on the poop deck under the scrutiny of the portmaster). New zombies are forbbidden entrance and industry blossoms!
On the mainland the problem is that there is a bottomless reservoir of zombies stumbling in from the benighted hinterlands and so every time they think things are under control - zombomania redux.