In a large pot, combine the damsons, apples, white sugar, brown sugar, white vinegar, apple cider vinegar, onion, garlic, ginger, mustard seeds, cinnamon, cloves, and nutmeg.
Bring the mixture to a boil over medium-high heat, then reduce the heat to medium-low and let it simmer for about 1 hour or until the fruit is soft and the mixture has thickened. Be sure to stir occasionally to prevent the chutney from sticking to the bottom of the pot.
Once the chutney has thickened, remove it from the heat and let it cool slightly.
Using an immersion blender or a regular blender, blend the chutney until it's smooth and free of chunks.
Pour the chutney into clean, sterilized jars and let it cool to room temperature. Once cooled, the chutney can be stored in the refrigerator for up to 3 months.
You can also can the chutney to preserve it for longer. To can the chutney, pour it into sterilized jars and process them in a boiling water bath for 10 minutes. Be sure to follow proper canning techniques and guidelines to ensure the chutney is safe to eat.