Cascade systems comprise a two-model sequence, with a lightweight model processing all samples and a heavier, higher-accuracy model conditionally refining harder samples to improve accuracy. By placing the light model on the device side and the heavy model on a server, model cascades constitute a widely used distributed inference approach. With the rapid expansion of intelligent indoor environments, such as smart homes, the new setting of Multi-Device Cascade is emerging where multiple and diverse devices are to simultaneously use a shared heavy model on the same server, typically located within or close to the consumer environment. This work presents MultiTASC, a multi-tenancy-aware scheduler that adaptively controls the forwarding decision functions of the devices in order to maximize the system throughput, while sustaining high accuracy and low latency. By explicitly considering device heterogeneity, our scheduler improves the latency service-level objective (SLO) satisfaction rate by 20-25 percentage points (pp) over state-of-the-art cascade methods in highly heterogeneous setups, while serving over 40 devices, showcasing its scalability.