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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 $R\,{=}\,2.37_{-0.12}^{+0.16}$ 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 $R\,{=}\,1.56_{-0.13}^{+0.15}$ 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 $13.5_{-9.0}^{+11.0}$ 13.5 − 9.0 + 11.0 M⊕ (<36.8 M⊕ at 2-σ) for TOI-1266 b and 2.2−1.5+2.0 $2.2_{-1.5}^{+2.0}$ 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 $0.09_{-0.05}^{+0.06}$ 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. [ABSTRACT FROM AUTHOR]