  We investigate the relations between nuclear star clusters (NSCs) and their
host galaxies, and between the structural properties of nucleated and
non-nucleated galaxies. We also address the environmental influences on the
nucleation of galaxies in the Fornax main cluster and the Fornax A group. We
select 557 Fornax galaxies ($10^{5.5} M_{\odot} < M_{\rm *,galaxy} < 10^{11.5}
M_{\odot} $) for which structural decomposition models and non-parametric
indices are available. We determine galaxy nucleation based on a combination of
visual inspection and a model selection statistic, the Bayesian information
criterion (BIC). We also test the BIC as an unsupervised method to determine
nucleation labels. We find a dichotomy in the properties of nuclei which reside
in galaxies more or less massive than $M_{\rm *,galaxy} \approx 10^{8.5}
M_{\odot}$. Specifically, the nuclei tend to be bluer than their host galaxies
and follow a scaling relation of $M_{\rm *,nuc} \propto {M_{\rm
*,galaxy}}^{0.5}$ for $M_{\rm *,galaxy} < 10^{8.5} M_{\odot}$. In galaxies with
$M_{\rm *,galaxy} > 10^{8.5} M_{\odot}$ we find that nuclei are redder compared
to the host and follow $M_{\rm *,nuc} \propto M_{\rm *,galaxy}$. Comparing
early-type galaxies, we find that nucleated galaxies tend to be redder in
global ($g'-r'$) colour, have redder outskirts relative to their own inner
regions ($\Delta (g'-r')$), be less asymmetric ($A$) and exhibit less scatter
in the brightest second order moment of light ($M_{20}$) than their
non-nucleated counterparts at a given stellar mass. Additionally, we find the
nucleation fractions to be typically higher in the Fornax main cluster than in
the Fornax A group, and that the nucleation fraction is highest towards the
centre of their respective environments. We also find that the BIC can recover
our labels of nucleation up to an accuracy of 97\%. (abridged)
