The present study performs a comprehensive fairness analysis of machine learning (ML) models for the diagnosis of Mild Cognitive Impairment (MCI) and Alzheimer's disease (AD) from MRI-derived neuroimaging features. Biases associated with age, race, and gender in a multi-cohort dataset, as well as the influence of proxy features encoding these sensitive attributes, are investigated. The reliability of various fairness definitions and metrics in the identification of such biases is also assessed. Based on the most appropriate fairness measures, a comparative analysis of widely used pre-processing, in-processing, and post-processing bias mitigation strategies is performed. Moreover, a novel composite measure is introduced to quantify the trade-off between fairness and performance by considering the F1-score and the equalized odds ratio, making it appropriate for medical diagnostic applications. The obtained results reveal the existence of biases related to age and race, while no significant gender bias is observed. The deployed mitigation strategies yield varying improvements in terms of fairness across the different sensitive attributes and studied subproblems. For race and gender, Reject Option Classification improves equalized odds by 46% and 57%, respectively, and achieves harmonic mean scores of 0.75 and 0.80 in the MCI versus AD subproblem, whereas for age, in the same subproblem, adversarial debiasing yields the highest equalized odds improvement of 40% with a harmonic mean score of 0.69. Insights are provided into how variations in AD neuropathology and risk factors, associated with demographic characteristics, influence model fairness.