  We explore how the characteristics of the cross-correlation functions between
the 21cm emission from the spin-flip transition of neutral hydrogen (HI) and
early Lyman-$\alpha$ (Ly$\alpha$) radiation emitting galaxies (Ly$\alpha$
emitters, LAEs) depend on the reionisation history and topology and the
simulated volume. For this purpose, we develop an analytic expression for the
21cm-LAE cross-correlation function and compare it to results derived from
different Astraeus and 21cmFAST reionisation simulations covering a physically
plausible range of scenarios where either low-mass ($<10^{9.5}M_\odot$) or
massive ($>10^{9.5}M_\odot$) galaxies drive reionisation. Our key findings are:
(i) the negative small-scale ($<2$ cMpc) cross-correlation amplitude scales
with the intergalactic medium's (IGM) average HI fraction
($\langle\chi_\mathrm{HI}\rangle$) and spin-temperature weighted overdensity in
neutral regions ($\langle1+\delta\rangle_\mathrm{HI}$); (ii) the inversion
point of the cross-correlation function traces the peak of the size
distribution of ionised regions around LAEs; (iii) the cross-correlation
amplitude at small scales is sensitive to the reionisation topology, with its
anti-correlation or correlation decreasing the stronger the ionising emissivity
of the underlying galaxy population is correlated to the cosmic web gas
distribution (i.e. the more low-mass galaxies drive reionisation); (iv) the
required simulation volume to not underpredict the 21cm-LAE anti-correlation
amplitude when the cross-correlation is derived via the cross-power spectrum
rises as the size of ionised regions and their variance increases. Our analytic
expression can serve two purposes: to test whether simulation volumes are
sufficiently large, and to act as a fitting function when cross-correlating
future 21cm signal Square Kilometre Array and LAE galaxy observations.
