Chromatin organization within the nuclear volume is essential to regulate many aspects of its function and to safeguard its integrity. A key player in this spatial scattering of chromosomes is the nuclear envelope (NE). The NE tethers large chromatin domains through interaction with the nuclear lamina and other associated proteins. This organization is perturbed in cells from Hutchinson-Gilford progeria syndrome (HGPS), a genetic disorder characterized by premature aging features. Here, we show that HGPS-related lamina defects trigger an altered 3D telomere organization with increased contact sites between telomeres and the nuclear lamina, and an altered telomeric chromatin state. The genome-wide replication timing signature of these cells is perturbed, with a shift to earlier replication for regions that normally replicate late. As a consequence, we detected a higher density of replication forks traveling simultaneously on DNA fibers, which relies on limiting cellular dNTP pools to support processive DNA synthesis. Remarkably, increasing dNTP levels in HGPS cells rescued fragile telomeres, and improved the replicative capacity of the cells. Our work highlights a functional connection between NE dysfunction and telomere homeostasis in the context of premature aging.