Heat the oven to 180 C / 350 F. Grease a 20 cm / 8-inch springform tin and line the base with parchment.
Peel, core, and dice the Bramley apples into 1 cm pieces. Set aside.
Cream the butter and caster sugar together in a large mixing bowl with an electric hand mixer for 3 to 4 minutes, until pale and fluffy.
Beat in the eggs one at a time, then mix in the lemon zest.
Sift the self-raising flour, baking powder, and salt over the batter. Fold in gently with a rubber spatula until just combined.
Fold in the diced apples, then add the milk a tablespoon at a time until the batter falls off the spatula in a thick ribbon.
Spoon the batter into the prepared tin and level the top. Scatter the demerara sugar evenly over the surface.
Bake for 55 to 65 minutes, until a skewer inserted in the center comes out clean and the top is deep golden. Tent loosely with foil after 40 minutes if it's browning too fast.
Cool the cake in the tin for 15 minutes, then release the springform sides and transfer to a wire rack to cool completely.
Slice and serve at room temperature, with a wedge of cheddar or a spoon of clotted cream if you like.