Preheat the oven to 200C/180C Fan/Gas 6.
For the pastry, pulse the flour and butter in a food processor until the mixture resembles breadcrumbs (or rub the ingredients together in a large mixing bowl using your fingertips).
Add the sugar, egg and 1-2 tablespoons of cold water and mix again until the mixture comes together as a smooth dough.
Turn out the dough onto a lightly floured work surface and roll it out into a circle that's about 3mm thick and large enough to line the tart tin. Press the pastry into the base and sides of the tin and make a small lip around the top with the excess pastry. Prick the base all over with a fork, then chill in the fridge for 15 minutes.
Line the chilled pastry case with baking paper and baking beans, then bake in the oven for 12-15 minutes, or until the lip of the pastry is light golden-brown. Remove the baking beans and paper, then reduce the oven temperature to 160C/140C Fan/Gas 3 and bake for a further 8-10 minutes, or until the pastry case is crisp and golden-brown all over. Set the pastry case aside in its tin to cool, leaving the oven on.
For the filling, place the apples, sugar and 2 tablespoons of water in a large, deep, lidded saucepan and cook over a medium heat, stirring gently, for 4-5 minutes. Cover the pan with the lid and bring the mixture to a gentle simmer. Simmer, without stirring, for 5-10 minutes, or until the apples have softened but still retain their shape.
Stir the blackberries into the cooked apples and continue to simmer for a further 5-10 minutes, or until all of the fruit is just tender. Set aside to cool.
Meanwhile, for the crumble topping, put the flour and butter into a large mixing bowl and rub the ingredients together using your fingertips, until the mixture resembles breadcrumbs. Add the remaining crumble topping ingredients and rub them into the mixture until small clumps have formed. (Alternatively, pulse all of the crumble topping ingredients in a food processor - there is no need to wash this out after making the pastry.) Set the crumble topping aside.
Suspend a colander over a bowl and strain the cooked fruits, collecting the juices. Spoon the fruits into the cooked pastry case. (If there is too much fruit to fit into the pastry case, add the remaining fruit to the reserved fruit juice.)
Sprinkle the crumble topping all over the fruit filling and bake the tart at 200C/180C Fan/Gas 4 for 20-25 minutes, or until the crumble is crisp and golden-brown.
Meanwhile, boil the reserved fruit juice in a saucepan for 3-4 minutes, or until thickened to a syrup.
Serve the tart warm, in slices, with the extra fruit juice spooned over it and either custard or cream on the side.