  Heavy neutral leptons are motivated by several extensions of the Standard
Model and their presence induces modifications in the lepton mixing matrix,
including new Dirac and Majorana CP violating phases. It has been recently
shown that these phases play an important role in lepton number and lepton
flavour violating decays and transitions, with a striking impact for the
predicted rates of certain observables. In this work, we now consider the
potential role of the heavy neutral fermions and of the new CP violating phases
in Higgs and $Z$-boson lepton flavour violating decays $H,Z\to
\ell_\alpha^\pm\ell_\beta^\mp$, as well as in the corresponding CP asymmetries.
In order to allow for a thorough analysis, we derive the full analytical
expressions of the latter observables, taking into account the effect of the
(final state) charged lepton masses. A comprehensive exploration of lepton
flavour violating $Z$ and Higgs decays confirms that these are very sensitive
to the presence of the additional heavy neutral leptons. Moreover, the new CPV
phases have a clear impact on the decay rates, leading to both destructive and
constructive interferences. Interestingly, in the $\mu\tau$ sector, the $Z\to
\ell_\alpha^\pm\ell_\beta^\mp$ rates are within reach of FCC-ee, with
associated CP asymmetries that can potentially reach up to $20\% - 30\%$.
