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Rapidly rotating, strongly magnetized neutron stars (millisecond proto-magnetars) formed in stellar core-collapse, neutron star mergers, and white dwarf accretion-induced collapse have long been proposed as central engines of gamma-ray bursts (GRB) and accompanying supernovae/kilonovae. However, during the first few seconds after birth, neutrino heating drives baryon-rich winds from the neutron star surface, potentially limiting the magnetization and achievable Lorentz factors of the outflow and casting doubt on whether proto-magnetars can launch ultra-relativistic jets at early times, as needed to power short-duration GRB. I present the first three-dimensional general-relativistic magnetohydrodynamic simulations of neutrino-heated proto-magnetar winds. While the global wind properties broadly agree with previous analytic estimates calibrated to one-dimensional models, our simulations reveal essential multidimensional effects. For rapidly rotating models with spin periods P ~ 1 ms, centrifugal forces strongly enhance mass loss near the rotational equator, producing a dense, sub-relativistic outflow (v ~ 0.1 c). This equatorial wind naturally confines and collimates less baryon-loaded outflows emerging from higher latitudes, leading to the formation of a structured bipolar jet with a peak magnetization along the pole up to ~30-100, sufficient to reach bulk Lorentz factors ~ 100 on larger scales. The resulting angular stratification of the outflow energy into ultra-relativistic polar and sub-relativistic equatorial components is broadly consistent with the observed partition between beaming-corrected GRB energies and supernova/kilonova ejecta. Our results demonstrate that millisecond proto-magnetars can launch relativistic jets within seconds of formation and highlight their potential role in powering the diverse electromagnetic counterparts of compact-object explosions.