  We use a semi-analytic model, ${\it Delphi}$, that jointly tracks the dark
matter and baryonic assembly of high-redshift ($z \simeq 4-20$) galaxies to
gain insight on the number density of Direct Collapse Black Hole (DCBH) hosts
in three different cosmologies: the standard Cold Dark Matter (CDM) model and
two Warm Dark Matter (WDM) models with particle masses of 3.5 and 1.5 keV.
Obtaining the Lyman-Werner (LW) luminosity of each galaxy from ${\it Delphi}$,
we use a clustering bias analysis to identify all, pristine halos with a virial
temperature $T_{vir}>=10^4$ K that are irradiated by a LW background above a
critical value as, DCBH hosts. In good agreement with previous studies, we find
the DCBH number density rises from $\sim10^{-6.1}$ to $\sim 10^{-3.5}\,
\mathrm{cMpc^{-3}}$ from $z\simeq 17.5$ to $8$ in the CDM model using a
critical LW background value of $30 J_{21}$ (where $J_{21}= 10^{-21} \, {\rm
erg\, s^{-1}\, Hz^{-1} \, cm^{-2} \, sr^{-1}}$). We find that a combination of
delayed structure formation and an accelerated assembly of galaxies results in
a later metal-enrichment and an accelerated build-up of the LW background in
the 1.5 keV WDM model, resulting in DCBH hosts persisting down to much lower
redshifts ($z \simeq 5$) as compared to CDM where DCBH hosts only exist down to
$z \simeq 8$. We end by showing how the expected colours in three different
bands of the Near Infrared Camera (NIRCam) onboard the forthcoming James Webb
Space Telescope (${\it JWST}$) can be used to hunt for potential $z \simeq 5-9$
DCBHs, allowing hints on the WDM particle mass.
