By the 16th and 17th centuries, Europe was gobbling up enormous amounts of wood. They were using it for their buildings, heating their houses, making ships, smelting iron, making glass, and all kinds of other things. Part of why steam power ever became a thing is that wood fuel prices were climbing higher and higher, and it started to make economic sense to push hard on coal mines. Many coal mines were below the water table, flooded, and the earliest steam engines were designed specifically for pumping the water out; the steam engine was hungry for water and coal, and those two things were right there if you had a flooded coal mine, needing no other infrastructure. Coal let England escape from the timber famine.
I've been trying to imagine what the opposite of steampunk might be, where coal deposits just weren't as large as they are on Earth, or where coal mines stalled out for some reason, and what it might have looked like for timber famine to sweep across Europe. There's be intensive forestry efforts, certainly, but also some innovations in fuel-savings forced by the heavy economic pressure, something that Europe was already going through.
There were already historical examples of forest fires set as a part of wartime tactics, or to otherwise control the supply of timber, even as late as the second World War. So that would intensify in this hypothetical lumberpunk setting, where most of the forests in Europe are carefully managed by forestry programs. Without coal and oil, every tank represents an enormous quantity of burned wood.
And while I don't think the economics would work out for transporting lumber across the ocean, the "untouched" forests of the New World would surely be a source of finished goods, because if you need lumber for ships, you can just build the ship in situ instead of putting the lumber onto a ship and making along voyage with it. And Russia, which is sitting on Siberian forests, can probably go much longer without being wood-starved than anywhere else, though I think it would need trains to transport the lumber east, and I'm not sure we'd have trains if all the iron had to be produced using charcoal, all the rail ties are lumber, and the trains themselves run on ... wood gas or something? I know you can convert a car to run on wood gas, something that was done during wartime when there were fuel shortages, but I think metal being semi-precious probably kills the dream of a car dead, even if I could see tanks working.
And I'm just now remembering about peat production, which maybe throws a wrench in things for various places, though it's obviously not a true substitute for coal unless you have huge reserves of the stuff.
