Cook the cranberries: Combine the cranberries, sugar and orange juice in a large saucepan. Bring to a boil, then reduce the heat to medium low; stir in the allspice, nutmeg and cinnamon stick and simmer 5 to 7 minutes.
Make the filling: Strain the cranberry mixture; set the berries aside. Return the liquid to the saucepan, bring to a simmer over medium-low heat and cook until syrupy, about 8 minutes.
Discard the cinnamon stick, then puree half of the cranberries in a blender; transfer to a bowl. Stir in the rest of the cranberries and the reduced syrup. Spread the mixture in a large baking dish so it cools more quickly. Refrigerate until completely cooled, about 20 minutes.
Make the batter: Preheat the oven to 375 degrees F. Sift the flour, baking powder and cream of tartar into a bowl. Combine the eggs, granulated sugar, lemon zest and 1/3 cup water in a stand mixer fitted with a whisk attachment. Beat on high speed until light and airy, 3 to 5 minutes. Gently fold the egg mixture into the flour mixture with a rubber spatula. Do not overmix: The batter will lose some volume, but small bubbles should remain.
Bake the cake: Butter a 10-by-15-inch jelly roll pan, line with parchment paper and butter the parchment. Add the batter. Slowly tilt the pan back and forth to evenly distribute the batter. Bake until the cake is golden brown, 12 to 15 minutes.
Unmold the cake: While the cake is still warm, place a slightly damp kitchen towel on top; lay a cutting board over the towel. Hold the cutting board and baking sheet together and flip over. Lift off the pan, then peel off the parchment. Let the cake cool slightly, 3 to 5 minutes.
Add the filling: Dollop the cooled filling over the cake and spread gently, leaving a 1-inch border on all sides. (Be careful not to tear the cake.) Sprinkle with the walnuts.
Roll the cake: Position the cake with a shorter end facing you. Use the towel to tip the edge of the cake forward and gently roll it over, using the towel to guide it. Once you make one full roll, pull the towel away and gently squeeze the cake to make the roll compact.
Continue rolling the cake, using the towel to guide it and gently squeezing as you go. Position the cake roll seam-side down and refrigerate at least 1 hour or overnight. Transfer to a platter and dust with confectioners' sugar. Slice and serve with whipped cream.
Chill the cake seam-side down to prevent it from unrolling.