  We study the thermodynamics of the spin-$S$ two-dimensional quantum
Heisenberg antiferromagnet on the square lattice with nearest ($J_1$) and
next-nearest ($J_2$) neighbor couplings in its collinear phase ($J_2/J_1>0.5$),
using the pure-quantum self-consistent harmonic approximation. Our results show
the persistence of a finite-temperature Ising phase transition for every value
of the spin, provided that the ratio $J_2/J_1$ is greater than a critical value
corresponding to the onset of collinear long-range order at zero temperature.
We also calculate the spin- and temperature-dependence of the collinear
susceptibility and correlation length, and we discuss our results in light of
the experiments on Li$_2$VOSiO$_4$ and related compounds.
