We calculate the energy-dependent cross section of the $np \leftrightarrow d\gamma$ process in chiral effective field theory and apply state-of-the-art tools for quantification of theory uncertainty. We focus on the low-energy regime, where the magnetic dipole and the electric dipole transitions cross over, including the range relevant for big-bang nucleosynthesis. Working with the leading one- and two-body electromagnetic currents, we study the order-by-order convergence of this observable in the chiral expansion of the nuclear potential. We find that the Gaussian process error model describes the observed convergence very well, allowing us to present Bayesian credible intervals for the truncation error with correlations between the cross sections at different energies taken into account. We obtain a 1$\sigma$ estimate of about 0.2\% for the uncertainty from the truncation of the nuclear potential. This is an important step towards calculations with statistically interpretable uncertainties for astrophysical reactions involving light nuclei.