Line the bottom of a 2- to 2 1/2-quart bowl with bread, tearing bread as necessary to completely cover bottom without overlapping. Halve remaining bread slices lengthwise. Line sides of bowl with overlapping bread slices (reserve some for covering the top); trim bread flush with top of bowl.
Combine berries, sugar, cinnamon, and 1/4 cup water in a large saucepan. Cook over medium-high heat, gently stirring a few times, just until berries start to soften and currants begin to burst and release their juices. Do not let mixture boil. Remove from heat. Let cool 10 minutes.
Pour berry mixture into bowl and cover with a single layer of remaining bread slices, tearing bread as necessary to fill in any gaps. Place a round dish or pan slightly smaller than the diameter of the bowl on top of bread, and press down on plate until juices rise to surface. Place a heavy can or other weight on top of plate. Transfer to refrigerator and let chill at least 12 hours and up to 1 day.
Remove weighted plate from basin. Run a thin knife around edge between bread and bowl. Invert onto a rimmed serving plate and lift bowl to release pudding. Stir together mascarpone and cream in a bowl until smooth, then whisk to soft peaks. Cut pudding into wedges and serve with whipped mascarpone cream.