  Using a heavy-mass effective field theory (HEFT), we study gravitational-wave
emission in the scattering of two spinless black holes or neutron stars of
arbitrary masses at next-to-leading order in the Post-Minkowskian expansion. We
compute the contributions to the one-loop scattering amplitude with four
scalars and one graviton which are relevant to the calculation of the
waveforms, also presenting expressions of classical tree-level amplitudes with
four scalars and up to two radiated gravitons. The latter are obtained using a
novel on-shell recursion relation for classical amplitudes with four scalars
and an arbitrary number of gravitons. Our one-loop five-point amplitude is
expressed in terms of a single family of master integrals with the principal
value prescription for linearised massive propagators, which we evaluate using
differential equations. In our HEFT approach all hyper-classical iterations and
quantum corrections to the amplitude are dropped at the diagrammatic level,
thereby computing directly contributions to classical physics. Our result
exhibits the expected factorisation of infrared divergences, the correct soft
limits, and highly nontrivial cancellations of spurious poles. Finally, using
our amplitude result we compute numerically the corresponding next-to-leading
corrections to the spectral waveforms and the far-field time-domain waveforms
using the Newman-Penrose scalar $\Psi_4$.
