What's your soil pH? Ashes tend to be very high in pH (remember reading about how ashes can be used to make lye for soap?). If your soil is acidic, you can spread the ashes on the garden and lawn to raise the pH. I'd be reluctant to use them here where the soil tends to be alkaline, but somebody in Colorado, where conditions are similar, has reported here that he spreads ashes on his lawn with no problems.

Composting is supposed to ameliorate the alkalinity, but I would not add a lot at one time for fear of overwhelming the compost.

Wood ash is of course very variable but a ball park N-P-K number is 0-1-3. Most western soils have adequate amounts of potassium. Wood ash as has been said, a liming effect. So unless you live east of the Mississippi or on the wet west coast, I would not add wood ash to my garden without a soil test.

Adding wood ash in the compost can be just as tricky. Nitrogen is the most volatile of the nutrients. As the pH increases, ammonium turns into ammonia gas and leaves the compost pile.

This doesn't give you many options, does it? You can take up pottery and use it as a glaze ingredient.

I did not have my soil tested last year. I was more concerned by the fact that it was clay. I was also too late in the season and out of funds to add amendments;( I am planning to add compost and some good soil to my garden area for some raised beds(mixed with the underlying compost), and will have my soil tested then.

Used in small doses, wood ashes can be used to suppress slugs in the garden. Since I live in acidic NC, I use wood ashes on the garden directly, but only in the winter when there's time for the rain to work it into the soil.

Whether ashes from wood stoves, pellet stoves, etc. are good for the garden is something that is often debated vigorously, and most knowledgable people will tell you it depends on how much and what was the source. Ashes can have nutrient levels you may need and they will be quite alkaline which could change your soils pH enough to make the nutrients in the ashes unavailable to the plants.
Before adding ashes to your garden have a good, reliable soil test done to see what those ashes might do, might contribute. If you have a soil pH above 6.4 you might not want to add any ash since that could raise the soils pH above 7.0 and very few plants like a soil pH that high and most nutrients become less readily available above that level.

Aside from the pH issue, the particle size of ashes is mostly very very small - clay size, in other words. It would not do any good for improving the particle size distribution of clay soil, that's for sure.