  The on-shell matrix elements, or couplings $g_{H H^*(H_1)\gamma}$, describing
the $B(D)_q^* \to B(D)_q \gamma$ and $B_{1 q} \to B_q \gamma$ ($q = u,d,s$)
radiative decays, are determined from light-cone sum rules at next-to-leading
order for the first time. Two different interpolating operators are used for
the vector meson, providing additional robustness to our results. For the
$D^*$-meson, where some rates are experimentally known, agreement is found. The
couplings are of additional interest as they govern the lowest pole residue in
the $B(D) \to \gamma$ form factors which in turn are connected to
QED-corrections in leptonic decays $B(D) \to \ell \bar \nu$. Since the
couplings and residues are related by the decay constants $f_{H^*(H_1)}$ and
$f^T_{H^*(H_1)}$ respectively, we determine them at next-leading order as a
by-product. The quantities $\{ f_{H^*}^T, f_{H_1},f_{H_1}^T\}$ have not
previously been subjected to a QCD sum rule determination. All results are
compared with the existing experimental and theoretical literature.
