As there is no way for us to know what was burned and so what is in your ash, it must go in your green household waste bin rather than composting via garden waste wheelie bins.
Other recycling options: If you burn wood, the ash is a great dry addition to a home compost bin. Find out about discounted bins at www.getcomposting.com. Wood ash can also be put on your flower bed (it’s a high potash fertiliser).