  We use modified linear spin-wave theory (MLSWT) to study ground-state
entanglement for a length-$L$ line subsystem in $L\times L$ square- and
triangular-lattice quantum Heisenberg antiferromagnets with coplanar spiral
magnetic order with ordering vector $\mathbf{Q}=(q,q)$ and $N_G=3$ Goldstone
modes, except if $q=\pi$ (collinear order, $N_G=2$). Generalizing earlier MLSWT
results for $q=\pi$ to commensurate spiral order with $s\geq 3$ sublattices
($q=2\pi r/s$ with $r$ and $s$ coprime), we find analytically for large $L$ a
universal and $n$-independent subleading term $(N_G/2)\ln L$ in the R\'{e}nyi
entropy $S_n$, associated with $L^{1/2}$ scaling of $\lambda_0$ and
$\lambda_{\pm q}$, with $\lambda_0\neq \lambda_{\pm q}$ for spiral order; here
$\{\lambda_{k_y}\}$ are the $L$ mode occupation numbers of the entanglement
Hamiltonian. The term $(3/2)\ln L$ in $S_n$ agrees with a nonlinear sigma model
(NLSM) study of $s=3$ spiral order ($q=2\pi/3$). These and other properties of
$S_n$ and $\lambda_{k_y}$ are explored numerically for an anisotropic
nearest-neighbor triangular-lattice model for which $q$ varies in the spiral
phase.
