While Large Language Models (LLMs) have shown strong performance on clinical text understanding, they struggle with longitudinal prediction tasks such as dementia prognosis, which require reasoning over complex, non-monotonic symptom trajectories across multiple visits. Standard supervised training lacks explicit annotations for symptom evolution, while direct Reinforcement Learning (RL) is hindered by sparse binary rewards. To address this challenge, we introduce Dementia-R1, an RL-based framework for longitudinal dementia prognosis from unstructured clinical notes. Our approach adopts a Cold-Start RL strategy that pre-trains the model to predict verifiable clinical indices extracted from patient histories, enhancing the capability to reason about disease progression before determining the final clinical status. Extensive experiments show that Dementia-R1 achieves the best overall performance on the AMC real-world unstructured cohort, reaching an AUROC of 84.02% and outperforming models up to 10x larger. The framework also generalizes to Parkinson's disease dementia prediction in an independent hospital cohort, achieving an AUROC of 78.37%. On the ADNI benchmark, our 7B model attains the highest AUROC among all LLM baselines at 83.17%, demonstrating strong longitudinal reasoning over fluctuating cognitive trajectories. Code is available at https://anonymous.4open.science/r/dementiar1-CDB5.