We have investigated the antiferromagnetic phase transition in the frustrated and multiferroic hexagonal manganites $h$-(Y$_{0.98}$ Eu$_{0.02}$)MnO$_3$ (YEMO) and $h$-YMnO$_3$ (YMO). Elastic neutron scattering has been used to study in detail the phase transition in YMO and YEMO in zero pressure and in YMO at a hydrostatic pressure of 1.5~GPa. In zero pressure, we find the critical temperature $T_{\rm N} = 72.11(5)$~K and 71.3(1)~K and the critical exponent $\beta = 0.206(3)$ and 0.22(2), for YEMO and YMO, respectively. This is in agreement with earlier work by Roessli {\em et al.}. In applied hydrostatic pressure of 1.5~GPa, the ordering temperature increases to $T_{\rm N} = 75.2(5)$~K, in agreement with earlier reports, while $\beta$ is unchanged. Inelastic neutron scattering was used to determine the anisotropy spin wave gap at the phase transition, expected from spin wave theory to close with a critical exponent $\beta'$ identical to the one of the order parameter, $\beta' = \beta$. Our results indicate that the gap in YEMO indeed closes at $T_{\rm N}=72.4(3)$~K with $\beta' = 0.24(2)$, while the in-pressure gap in YMO closes at 75.2(5)~K with an exponent of $\beta' =0.19(3)$. In addition, the low-temperature anisotropy gap was found to have a slightly higher absolute value under pressure. The consistent values obtained for $\beta$ in the two systems support the likelihood of a new universality class for triangular, frustrated antiferromagnets.