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We present a multi-epoch test of f(T) gravity with nonminimal torsion-matter coupling, combining early- and late-Universe observations. At the MeV scale, Big-Bang Nucleosynthesis constrains the fractional variation of the weak freeze-out temperature, |δτ_f/τ_f|, thereby mapping light-element abundances into limits on deviations from the standard expansion history. At low redshift, we confront the model with type Ia supernovae, baryon acoustic oscillations, and cosmic-chronometer data, which respectively probe distances, the late-time standard ruler, and the Hubble rate. Independent analyses highlight the complementary roles of each dataset, while a joint SNe Ia + BAO + CC fit breaks degeneracies and yields the tightest combined bounds. As an illustration, we examine two representative torsion-modified gravity scenarios: BBN strongly limits large departures from standard cosmology, whereas late-time probes remain compatible with a near-ΛCDM background. This unified approach demonstrates the power of linking early-Universe nuclear physics with precision cosmological observables in assessing torsional extensions of gravity.