Mash blackberries in a saucepan with a potato masher. Stir in sugar until juices form; place about 1 tablespoon blackberry juice in a small bowl and stir in cornflour. Pour cornflour mixture into saucepan.
Bring berries to the boil, stirring often, until jam is thickened, about 15 minutes. Stir in cinnamon and allspice. Remove from heat and allow to cool. Transfer jam to a bowl, cover and refrigerate until chilled. Stir in lemon juice.