In recent years, there has been growing interest in the use of mobile cloud and Internet of Medical Things (IoMT) in automated diagnosis and health monitoring. These applications play a significant role in providing smart medical services in modern healthcare systems. In this paper, we deploy a mobile cloud-based IoMT scheme to monitor the progression of a neurological disorder using a test of motor coordination. The computing and storage capabilities of cloud server is employed to facilitate the estimation of the severity levels given by an established quantitative assessment. An Android application is used for data acquisition and communication with the cloud. Further, we integrate the proposed system with a data sharing framework in a blockchain network as an innovative solution that allows reliable data exchange among healthcare users. The experimental results show the feasibility of implementing the proposed system in a wide range of healthcare applications.