  We present a systematic study of leptogenesis in neutrino mass models with
$\mu\tau$-flavoured CP symmetry. In addition to the strong hierarchical
$N_1$-dominated scenario ($N_1$DS) in the `two flavour regime' of leptogenesis,
we show that one may choose the right-handed (RH) neutrino mass hierarchy as
mild as $M_2\simeq 4.7 M_1$ for a perfectly valid hierarchical $N_1$DS. This in
turn reduces the lower bound on the allowed values of $M_1$, compared to what
is stated in the literature. The consideration of flavour effects due to the
heavy neutrinos also translate into an upper bound on $M_1$. It is only below
this bound that the observed baryon-to-photon ratio can be realized for a
standard ${ N_1}$ domination, else a substantial part of the parameter space is
also compatible with $N_2$DS. We deduce conditions under which the baryon
asymmetry produced by the second RH neutrino plays an important role. Finally,
we discuss another interesting scenario where lepton asymmetry generated by
$N_2$ in the two flavour regime faces washout by $N_1$ in the three flavour
regime. Considering a hierarchical light neutrino mass spectrum, which is now
favoured by cosmological observations, we show that at the end of
$N_1$-leptogenesis, the asymmetry generated by $N_2$ survives only in the
electron flavour and around $33\%$ of the parameter space is consistent with a
pure $N_2$-leptogenesis.
