Delayed healing of diabetic wounds poses a major challenge to human health due to severe vascular dysfunction, sustained inflammation, and vulnerability to microbial infection. Herein, we constructed multidimensionally nano-topologized electrospun polycaprolactone (PCL) fibrous membranes with shish-kebab nanoarrays on each fiber through self-induced crystallization, on which the CuO2–MgO2 bimetallic peroxide nanodots (BPNs) were anchored by polydopamine (PDA) as the bridging layer. When activated by the acidic microenvironment (typically infected diabetic wound), BPNs on fibers reacted immediately to release Cu2+ and Mg2+ ions together with hydrogen peroxide (H2O2) molecules, which were then transferred into ·OH radicals through Fenton-type reactions catalyzed by Cu2+ for instant bacteria elimination. At the same time, the released Cu2+ and Mg2+ ions were retained to improve the angiogenesis and suppress the inflammation infiltration, thus remodeling the wound microenvironment. Meanwhile, the one-dimensional (1D)-constructed nano shish-kebabs and PDA coating on fibers provided additional topological activation for cell adhesion and directed migration along the aligned fiber orientation. Through the meticulous design, the resultant membranes markedly accelerated the infected wound healing in the diabetic rat model. This study pioneers a unique design to develop a nanocomposite fibrous membrane that combines multidimensional topologies with chemodynamic therapy (CDT), for efficiently combating infected diabetic wounds.