We report the discovery and characterisation of a super-Earth and a sub-Neptune transiting the bright (K = 8.8), quiet, and nearby (37 pc) M3V dwarf TOI-1266. We validate the planetary nature of TOI-1266 b and c using four sectors of TESS photometry and data from the newly-commissioned 1-m SAINT-EX telescope located in San Pedro Mártir (México). We also include additional ground-based follow-up photometry as well as high-resolution spectroscopy and high-angular imaging observations. The inner, larger planet has a radius of R = 2.37-0.12+0.16 R⊕ and an orbital period of 10.9 days. The outer, smaller planet has a radius of R = 1.56-0.13+0.15 R⊕ on an 18.8-day orbit. The data are found to be consistent with circular, co-planar and stable orbits that are weakly influenced by the 2:1 mean motion resonance. Our TTV analysis of the combined dataset enables model-independent constraints on the masses and eccentricities of the planets. We find planetary masses of Mp = 13.5-9.0+11.0 M⊕ (<36.8 M⊕ at 2-σ) for TOI-1266 b and 2.2-1.5+2.0 M⊕ (<5.7 M⊕ at 2-σ) for TOI-1266 c. We find small but non-zero orbital eccentricities of 0.09-0.05+0.06 (<0.21 at 2-σ) for TOI-1266 b and 0.04 ± 0.03 (< 0.10 at 2-σ) for TOI-1266 c. The equilibrium temperatures of both planets are of 413 ± 20 and 344 ± 16 K, respectively, assuming a null Bond albedo and uniform heat redistribution from the day-side to the night-side hemisphere. The host brightness and negligible activity combined with the planetary system architecture and favourable planet-to-star radii ratios makes TOI-1266 an exquisite system for a detailed characterisation.