Alzheimer's disease is characterized by the progressive accumulation of amyloid-$β$ and tau followed years later by cognitive impairment. Despite this established motif, substantial subject-level variability exists in the age of pathological progression and the onset of cognitive symptoms. To understand the source of this variation, subjects must be aligned across heterogeneous disease timelines via frameworks that jointly model disease progression and time to cognitive impairment with reference to landmark positivity thresholds. Existing neurodegenerative disease progression models rely on restrictive parametric forms, fail to anchor disease timelines to positivity thresholds, and decouple biomarker trajectories from cognitive survival endpoints. To address these limitations, we introduce the Bayesian Threshold-Aligned Joint Disease Progression Model (B-TAJ DPM). This generative, semi-parametric framework models multivariate disease progression trajectories over latent disease timelines anchored at landmark positivity thresholds. Crucially, the framework integrates a survival model to link pathological progression to cognitive impairment. Posterior inference and posterior predictions for unseen subjects are carried out in open-source software. Simulation studies demonstrate excellent estimation accuracy and interval coverage. When applied to Alzheimer's Disease Neuroimaging Initiative data, B-TAJ DPM characterizes non-linear progression patterns, quantifies subject-level variation in positivity age, and reveals links between age of tau positivity and acceleration of cognitive impairment.

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