In this classic English dessert, layers of bread are saturated with berry juice; it is important to use a firm, good-quality white sandwich bread.
Combine strawberries and sugar in a large straight-sided skillet over medium heat. Cook until berries release their juices and sugar dissolves, about 5 minutes. Add blueberries, raspberries, and blackberries, and cook until they release their juices but still hold their shape, about 5 minutes; the mixture will look like a thick soup. Remove from heat, and transfer to a mixing bowl to cool. Stir in lemon juice and salt.
Line six 1/2-cup ramekins with plastic wrap. Cut a circle to the diameter of a ramekin from each slice bread.
Spoon 2 tablespoons of the berry mixture into each ramekin. Dip six circles of bread into the bowl of berries, one at a time, saturating them; place one in each ramekin. Cover with 2 tablespoons more of the berry mixture. Continue until you have three layers of bread and berries, ending with bread. If there is not enough juice to saturate all the bread, press remaining berries through a coarse sieve to yield more juice. When complete, bread and berries should be brimming over the tops of the ramekins.
Arrange ramekins in a shallow baking pan, cover with plastic wrap, and place a board on top large enough to cover them all. Weight the board with heavy cans (about 5 pounds), and refrigerate overnight.
To serve, combine cream, vanilla, and confectioners' sugar in a chilled metal bowl; whip until stiff but soft, 2 to 3 minutes. Remove top layer of plastic wrap, and invert puddings onto plates; remove ramekins. Serve with a dollop of whipped cream.