Alzheimer's disease (AD) is a neurodegenerative condition that causes cognitive impairment and structural brain changes, and early diagnosis is crucial for treatment. This paper proposes a cloud-based hybrid deep learning architecture for scalable Alzheimer's disease staging based on structural magnetic resonance imaging (MRI). The hybrid architecture leverages convolutional neural networks (CNNs), attention, and Transformer-based modeling to capture both local and global spatial features in MRI images. CNN extracts multi-dimensional features, while the attention and Transformer layers enrich context representation for more accurate staging of the disease. The framework was trained on a GPU-enabled computer using the PyTorch deep learning framework on a publicly available MRI dataset for Alzheimer's disease from the Kaggle website. The model was cross-validated on the OASIS dataset to assess its generalizability. The experimental findings showed that the proposed approach achieved an accuracy of 99.65% on the Kaggle dataset, which is better than several state-of-the-art deep learning models such as VGG16, ResNet variants, DenseNet121, EfficientNet-B0, and Vision Transformer. This approach also achieved an accuracy of 91.67% on the OASIS dataset, demonstrating the model's ability to generalize across neuroimaging datasets. To enable practical usage and accessibility, the trained model was also deployed in a cloud-based inference environment on Hugging Face Spaces, which allows MRI image upload and prediction from a web browser. The scalable and cloud-deployable design allows scalable medical image processing and integration with cloud-based diagnostic systems as well as telemedicine-based healthcare solutions. These findings demonstrate the potential for robust and accessible staging of Alzheimer's disease using the proposed framework in clinical settings.