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We have considered the prospects for measuring the cross Warm Dark Matter(WDM) power spectrum of the redshifted HI 21-cm signal and the Lyman-$\alpha$forest and thereby constraining WDM mass using observations with upcomingradio-interferometers - the Ooty Wide Field Array (OWFA) and SKA1-mid, and aspectroscopic survey of the quasars. We have considered a quasar survey with amean observed quasar number density of $\bar{{\rm n}}_{\rm Q} = 48$ deg$^{-2}$over a collecting area of 14455 deg$^{2}$, and a mean spectroscopic SNR = 5.Our analysis with OWFA shows that it is possible to measure the WDM powerspectrum in several $k$-bins at $k < 0.4$ Mpc$^{-1}$ with ${\rm SNR} > 5$ usingan observation of 200 hours each in 100 different fields-of-view for $m_{\rmWDM} = 0.25$ keV. Considering the possibility of the joint measurement of theparameters, the warm dark matter density parameter $\Omega_{{\rm WDM}}$, andthe dark energy density parameter $\Omega_{\Lambda0}$, we find that therelative error on the $1-\sigma$ measurement of the parameter $\Omega_{{\rmWDM}}$ is $\sim 0.8$ for a fiducial $m_{\rm WDM} = 0.25$ keV. We further findthat it is possible to have a measurement of the suppression of power from theCold Dark Matter (CDM) power spectrum at a confidence level of $\sim7.2-\sigma$ and $\sim 2.7-\sigma$ in two different $k$-bins over the $k$-range$0.1 \leq k \leq 3.13$ Mpc$^{-1}$ for $m_{\rm WDM} = 0.15$ keV. Considering theanalysis with SKA1-mid, we find that for a fiducial $m_{\rm WDM}= 0.25$ keV,the suppression in the cross power spectrum can be measured at $\sim 10 -\sigma$ around $k \sim 0.2 \rm{Mpc}^{-1}$ for a total observing time of $20000$hrs distributed uniformly over $50$ independent pointings where the available$k$-range is binned as $\Delta k = k/5$.

Constraining warm dark matter power spectrum using the cross-correlation of HI 21 cm signal and the Lyman-$\alpha$ forest