New and innovative wearable IoT devices for health monitoring systems (HMS) have been invented one after another. However, most of these devices are resource-constrained with restricted energy and computation power. The HMS data need to be processed via mobile edge computing (MEC) to improve the response time to fulfill the latency-sensitive and computation-intensive applications and to reduce bandwidth consumption. This paper presents an efficient task scheduling and resource allocation mechanism in MEC to meet these demands in contemplating emergency conditions under HMS. We propose a priority-based task-scheduling and resource-allocation (PTS-RA) mechanism that can assign different priorities to different tasks by considering the tasks' emergency levels computed with respect to the data aggregated from a patient's smart wearable devices. The mechanism can optimally determine whether a task should be processed locally at the hospital workstations (HW) or in the cloud. This is aimed to reduce the total task processing time and the bandwidth cost as much as possible. The proposed approach is to ensure that tasks related to the emergency are given higher priorities and to run first. After the tasks� computations, results are sent to the doctor to response promptly with quick decisions. The proposed PTS-RA was benchmarked against state-of-the-art algorithms concerning average latency, task scheduling efficiency, task execution time, network usage, CPU utilization, and energy consumption. The benchmarking results are promising as PTS-RA is capable to manage the emergency conditions and is meeting the latency-sensitive tasks' requirements with reduced bandwidth cost. © 2023 The Authors