Big-bang nucleosynthesis (BBN) relates key cosmological parameters to the primordial abundance of light elements. In this paper, we point out that the recent observations of cosmic microwave background anisotropies by the Planck satellite and by the BICEP2 experiment constrain these parameters with such a high level of accuracy that the primordial deuterium abundance can be inferred with remarkable precision. For a given cosmological model, one can obtain independent information on nuclear processes in the energy range relevant for BBN, which determine the eventual H-2/H yield. In particular, assuming the standard cosmological model, we show that a combined analysis of Planck data and of recent deuterium abundance measurements in metal-poor damped Lyman-alpha systems provides independent information on the cross section of the radiative capture reaction d(p, gamma)(3) He converting deuterium into helium. Interestingly, the result is higher than the values suggested by a fit of present experimental data in the BBN energy range (10-300 keV), whereas it is in better agreement with ab initio theoretical calculations, based on models for the nuclear electromagnetic current derived from realistic interactions. Due to the correlation between the rate of the above nuclear process and the effective number of neutrinos N-eff, the same analysis points out a N-eff > 3 as well. We show how this observation changes when assuming a nonminimal cosmological scenario. We conclude that further data on the d(p, gamma)(3) He cross section in the few hundred keV range, which can be collected by experiments like LUNA, may either confirm the low value of this rate, or rather give some hint in favor of next-to-minimal cosmological scenarios.