limits of performance
\$e^{-\tau s} \rightarrow e^{-j\omega \tau}=cos(\omega \tau)-jsin(\omega \tau) = 1\angle( -\omega \tau)  \$

This means unity gain, and a phase lag proportional to angular frequency. At the cut-off frequency of the other the component of the TF, the phase angle contributed by the delay will be found by setting \$\omega =\omega_c\$, thus: \$\phi _{delay}=-\omega_c \tau\$ rad.