The deployment of Federated Learning (FL) across the Internet of Medical Things (IoMT) is severely hindered by computational asymmetry and statistical heterogeneity. Traditional synchronous aggregation protocols suffer from severe straggler effects when deployed across devices with varying computational capacities, such as hospital servers vs. ambulatory wearables. In this study, we propose the Asynchronous Proximal Federated Aggregation (APFA) framework to address these dual bottlenecks. APFA integrates a local proximal regularizer with a server-side staleness dampening penalty, permitting continuous, uncoordinated model updates from edge devices. Evaluated on highly skewed partitions of the CheXpert and MIMIC-IV datasets, APFA reached an 80% diagnostic viability threshold in just 4.1 simulated hours, representing a 71% reduction in total wait time compared to standard synchronous baselines like FedProx. Our mathematical integration effectively mitigates weight divergence, indicating the robustness of asynchronous machine learning for scalable, privacy-preserving clinical diagnostics.