Previously, spent wood ashes from making own lye were disposed of into the compost pile.
After digging into the new book, The Humanure Handbook, no substance that significantly alters pH including liming agents (eg: agricultural lime, wood ashes; sulphur on the other end of the pH scale) should be added to the pile. Instead, these substance should be added directly to the soil and can be added at the same time as compost. The issue lies in how the pH affects the microbes responsible for the process of active composting. Thus I would see no problem with ashes being added to entirely mature compost once the biological processes have terminated.
In fact, the book presents a finding that liming agents as well as other soil amendments like acidifiers and fertilizers are more fully utilized and reach deeper into soils that have been dressed with compost compared to untreated soils. The author asserts that increased organic matter is like responsible for the observations.
Composting any organic scraps from paper to veggie peelings to bones (and yes, human manure but we can take steps one at a time!) is an easy way one can take to lessen their environmental impact and keep precious nutrients from be lost forever to the anaerobic landfill. That compost can help any plant from lawns to veggies to potted houseplants to landscaping plants and everything in between!