Wood ash doesn't exactly compost as all the compostable organic stuff was already burned out of it.
Much wood ash contains calcium carbonate as its major component, representing 25 or even 45 percent. Less than 10 percent is potash, and less than 1 percent phosphate; there are trace elements of iron, manganese, zinc, copper and some heavy metals. However, these numbers vary, as combustion temperature is an important variable in determining wood ash composition. All of these are, primarily, in the form of oxides.
Note that none of it is Nitrogen.
[back to Wiki] Wood ash can be used as an organic fertilizer used to enrich soil nutrition. In this role, wood ash serves a source of potassium and calcium carbonate, the latter acting as a liming agent to neutralize acidic soils.
So with all that CaCO3 wood ash is strongly alkaline. That can be a good thing if your soil is too acid. Personally I've never lived anywhere where the soil was too acid. My current soil is slightly acid, but that is what most veggies prefer. If your soil is not too acid, you would need to be very careful with the wood ash.