  The Milky Way's nuclear stellar disc (NSD) and nuclear star cluster (NSC) are
the main features of the Galactic centre. Nevertheless, their observation is
hampered by the extreme source crowding and high extinction. Hence, their
relation and formation scenario are not fully clear yet. We aim at detecting
the stellar populations from the NSC and the NSD along the line-of-sight
towards the NSC, and assess whether they have different stellar populations and
star formation histories. We analysed the colour-magnitude diagram, $K_s$ vs.
$H-K_s$, of a region of $8.2'\times2.8'$ centred on the NSC, and detected two
different stellar groups with different extinctions. We studied their red
clumps to find the features associated to each of the stellar populations. We
obtained that the two groups of stars correspond to the NSD and the NSC, and
found that they have significantly different stellar populations and star
formation histories. We detected a double red clump for the NSD population, in
agreement with previous work, whereas the NSC presents a more complex structure
well fitted by three Gaussian features. We created extinction maps to analyse
the extinction variation between the detected stellar groups. We found that the
high-extinction layer varies on smaller scales (arc-seconds), and that there is
a difference of $A_{K_s}\sim0.6$\,mag between both extinction layers. Finally,
we obtained that the distance towards each of the stellar populations is
compatible with the Galactic centre distance, and found some evidence of a
slightly closer distance for the NSD stars ($\sim360\pm200$\,pc).
