Heat the oven to 350 F. Combine the graham cracker crumbs, melted butter and granulated sugar, then press the mixture firmly into the bottom of a 9-inch springform pan and bake for 9 minutes.
Reduce the oven to 325 F. Beat the cream cheese and sugar on low speed until smooth, then mix in the eggs one at a time. Blend in the sour cream, vanilla, lemon juice and salt without overbeating.
Pour the filling over the crust and place the springform pan in a larger roasting pan. Add enough hot water to the roasting pan to reach halfway up the springform pan.
Bake for 55 to 65 minutes, until the outer edge is set and the center still has a gentle wobble. Turn off the oven, open the door slightly and leave the cheesecake inside for 45 minutes.
Remove the cheesecake from the water bath and cool completely. Cover and refrigerate for at least 6 hours.
Cook the peaches, brown sugar, cornstarch, lemon juice, cinnamon and nutmeg over medium heat for 5 to 7 minutes, stirring gently, until glossy and thickened. Transfer the peach topping to a bowl and cool completely.
Combine the flour, oats, brown sugar, cinnamon, salt and melted butter. Spread the crumbs on a parchment-lined baking sheet and bake at 350 F for 14 to 17 minutes, stirring once, then cool.
Spoon the chilled peach topping over the cheesecake shortly before serving. Cover the peaches generously with the crisp crumble.