Healthcare has become exceptionally sophisticated, as wearables and connected medical devices revolutionize remote patient monitoring, emergency response, medication management, diagnosis, and predictive and prescriptive analytics. Internet of Things and Cloud computing integrated systems (IoT-Cloud) facilitate sensing, automation, and processing for these healthcare applications. While real-time response is crucial for alleviating patient emergencies, protecting patient privacy is paramount in data-driven healthcare. In this paper, we propose a multi-layer IoT, Edge, and Cloud architecture to enhance emergency healthcare response times by distributing tasks based on response criticality and data permanence requirements. We ensure patient privacy through a Differential Privacy framework applied across several machine learning models: K-means, Logistic Regression, Random Forest, and Naive Bayes. We establish a comprehensive threat model identifying three adversary classes and evaluate Laplace, Gaussian, and hybrid noise mechanisms across varying privacy budgets, with supervised algorithms achieving up to 83.6% accuracy. The proposed hybrid Laplace-Gaussian noise mechanism with adaptive budget allocation provides a balanced approach, offering moderate tails and better privacy-utility trade-offs for both low and high-dimension datasets. At the practical threshold of $\varepsilon$=5.0, supervised algorithms achieve 80-81% accuracy while reducing attribute inference attacks by up to 18% and data reconstruction correlation by 70%. We further enhance security through Blockchain integration, which ensures trusted communication through time-stamping, traceability, and immutability for analytics applications. Edge computing demonstrates 8$\times$ latency reduction for emergency scenarios, validating the hierarchical architecture for time-critical operations.