  The Milky Way centre exhibits an intense flux in the gamma and X-ray bands,
whose origin is partly ascribed to the possible presence of a large population
of millisecond pulsars (MSPs) and cataclysmic variables (CVs), respectively.
However, the number of sources required to generate such an excess is much
larger than what is expected from in situ star formation and evolution, opening
a series of questions about the formation history of the Galactic nucleus. In
this paper we make use of direct $N$-body simulations to investigate whether
these sources could have been brought to the Galactic centre by a population of
star clusters that underwent orbital decay and formed the Galactic nuclear star
cluster (NSC). Our results suggest that the gamma ray emission is compatible
with a population of MSPs that were mass segregated in their parent clusters,
while the X-ray emission is consistent with a population of CVs born via
dynamical interactions in dense star clusters. Combining observations with our
modelling, we explore how the observed $\gamma$ ray flux can be related to
different NSC formation scenarios. Finally, we show that the high-energy
emission coming from the galactic central regions can be used to detect black
holes heavier than $10^5\Ms$ in nearby dwarf galaxies.
