Effects of potassium hydroxide & carbonate on metals?
Your conjecture is at least partially correct, in that there was a mix of $\ce{KOH}$ and $\ce{K2CO3}$. However, there is only ~400 parts per million (ppm) $\ce{CO2}$ in air, so at first, the $\ce{KOH}$ was free to react with metal (and, possibly, the green lacquer on the PC board), until enough $\ce{CO2}$ entered the cabinet to form the carbonate. It is possible some raw $\ce{KOH}$ is still present, under a film of carbonate, so be careful. Note that there is ventilation in a PC cabinet, so $\ce{CO2}$ is never really depleted in it.

Far more damage, though, may be due to electrolytic corrosion. Wikipedia has a photo similar to yours, showing "Severe PCB corrosion from a leaky PCB mounted Ni-Cd battery". The liquid electrolyte creates bridges across the bare IC pins and across the PC board traces, and the continual current from the power supply and the not-quite-dead battery cause metal to be removed from one place and deposited in another.

After removing the battery, rinsing many times with distilled water, and tooth-brushing the board, you may find that surface traces need to be rebuilt (such as by soldering a bit of braid across a gap). This is likely a multilayer board, and if an internal trace is gone, it would be difficult to find and bridge the gap. Note that any such patches will affect timing, due to the impedance mismatch, so you may find that after all repairs, crashes occur with some frequency.

Best wishes for your restoration project. It's nice to know that at least one venerable computer is not going to landfill!