Advancements in networking technologies have enabled doctors to remotely diagnose and monitor patients using the Internet of Medical Things (IoMT), telemedicine, and edge-enabled healthcare. In e-healthcare, medical reports and patient records are typically outsourced to a server, which can make them vulnerable to unauthorized access and tampering. Therefore, it is crucial to ensure the authorization, security, confidentiality, and integrity of medical data. To address these challenges, this paper proposes a novel reversible watermarking approach with a high payload and low computational cost. First, the input medical image is divided into a Border region (BR) and a Non-Border region (NBR). The NBR region is upscaled using Neighbour Mean Interpolation (NMI) to ensure reversibility. The Electronic Patient Record (EPR) is encrypted using a pseudorandom key, which is generated adaptively from the host medical image and the Enigma machine. The encrypted EPR is then embedded in the medical image using NMI. Two levels of tamper detection (global and local) are performed at the receiver's end for higher accuracy. A Global Integrity Code is generated and embedded in BR using LSB embedding technique for global tamper detection. The experimental results show that the visual quality and robustness are both high (Avg. PSNR = 41.03 dB and Avg. SSIM = 0.99, NC = 0.99, and BER = 0.0019 calculated for 100 images). The subjective and objective experimental analysis indicates that the proposed scheme is highly secure and the computational cost is also low. The average embedding and extraction time (including embedding, encryption and decryption, extraction process respectively) is 0.88 s and 0.83 s. It is resistant to various image processing attacks. A comparison with some of the most recent popular schemes confirms the scheme's effectiveness.