In a medium saucepan, toss the sliced nectarines with 3/4 cup of the sugar and 2 tablespoons of the lemon juice and let stand until juicy, about 20 minutes. Bring the nectarines to a boil and simmer over low heat, mashing lightly, just until softened, about 8 minutes. Let cool.
Transfer the nectarines to a food processor and pulse to a coarse puree. Strain the puree, pressing hard on the skins to extract as much juice as possible. Add 2 tablespoons of the ginger liqueur and 1/2 cup of water to the puree; you should have about 3 cups of loose puree. Fill eight 6-ounce popsicle molds with half of the nectarine puree and freeze until firm but not solid, about 1 hour.
Meanwhile, in a bowl, combine the buttermilk with the remaining 1/4 cup of sugar, 1/2 tablespoon of lemon juice, 1 tablespoon of ginger liqueur and the lemon zest. Whisk until the sugar is dissolved, then refrigerate.
Pour the buttermilk mixture into the popsicle molds and freeze for about 1 hour, until firm but not solid. Pour the remaining nectarine puree into the molds and add the sticks, pushing them nearly through the popsicles. Freeze until firm, at least 4 hours and preferably overnight.
Dip the molds in hot water for a few seconds, then unmold the popsicles and serve right away.