  Nuclear electromagnetic currents are derived in time-ordered perturbation
theory within an effective-field-theory framework including explicit nucleons,
$\Delta$ isobars, and pions up to one loop, or N$^3$LO. The currents obtained
at N$^2$LO, {\it i.e.} ignoring loop corrections, are used in a study of
neutron radiative captures on protons and deuterons at thermal energies, and of
$A$=2 and 3 nuclei magnetic moments. The wave functions for $A$=2 are derived
from solutions of the Schr\"odinger equation with the Argonne $v_{18}$ (AV18)
or CD-Bonn (CDB) potentials, while those for $A$=3 are obtained with the
hyperspherical-harmonics-expansion method from a realistic Hamiltonian
including, in addition to the AV18 or CDB two-nucleon, also a three-nucleon
potential. With the strengths of the $\Delta$-excitation currents occurring at
N$^2$LO determined to reproduce the $n$-$p$ cross section and isovector
combination of the trinucleon magnetic moments, we find that the cross section
and photon circular polarization parameter, measured in $n$-$d$ and
$\vec{n}$-$d$ processes, are underpredicted by theory, for example the cross
section by (11--38)% as the cutoff is increased from 500 to 800 MeV. A complete
analysis of the results, in particular their large cutoff dependence, is
presented.
