Hutchinson-Gilford Progeria Syndrome (HGPS) is a premature aging disorder that causes severe cardiovascular disease, resulting in the death of patients in their teenage years. The disease pathology is caused by the accumulation of progerin, a mutated form of the nuclear lamina protein, lamin A. Progerin binds to the inner nuclear membrane, disrupting nuclear integrity, and causes severe nuclear abnormalities and changes in gene expression. This results in increased cellular inflammation, senescence, and overall dysfunction. The molecular mechanisms by which progerin induces the disease pathology are not fully understood. Progerin’s detrimental impact on nuclear mechanics and the role of the nucleus as a mechanosensor suggests dysfunctional mechanotransduction could play a role in HGPS. This is especially relevant in cells exposed to dynamic, continuous mechanical stimuli, like those of the vasculature. The endothelial (ECs) and smooth muscle cells (SMCs) within arteries rely on physical forces produced by blood flow to maintain function and homeostasis. Certain regions within arteries produce disturbed flow, leading to an impaired transduction of mechanical signals, and a reduction in cellular function, which also occurs in HGPS. In this review, we discuss the mechanics of nuclear mechanotransduction, how this is disrupted in HGPS, and what effect this has on cell health and function. We also address healthy responses of ECs and SMCs to physiological mechanical stimuli and how these responses are impaired by progerin accumulation.