  The redshifted $21$\,cm signal of neutral hydrogen can be used as a direct
probe of the intergalactic medium during Cosmic Dawn\,(CD) and Epoch of
Reionization\,(EoR). However, detecting this inherently weak signal has
numerous challenges. The major ones include accurate foreground removal from
low-frequency radio observations and systematics arising from instrumental
effects. The Earth's ionosphere poses a major obstacle at these low radio
frequencies. Thus, a systematic study of ionospheric effects on these sensitive
low-frequency observations is critical, given that the construction of the
Square Kilometre Array (SKA1-Low) is in full progress. We use the end-to-end
pipeline, called {\sc 21cme2e}, to study the effect of time-varying ionospheric
corruption on the $21$\,cm power spectrum recovery. We use two models: a) a
catalogue-based model focused on source position shift due to the refractive
effect of the ionosphere and b) a realistic ionospheric condition generated
using Kolmogorov's turbulence model. We assess the effect of the imperfections
thus generated on the extraction of \HI\ $21$\,cm signal power spectrum. Our
study shows that beyond ``median ionospheric offset" ($\theta_{\text{MIO}}
\lesssim 0.1''$), the $21$\,cm signal from the EoR is unaffected by residual
ionospheric effects. Our study emphasizes the need for the development of
efficient ionospheric calibration algorithms for the upcoming SKA1-Low
observations to extract the \HI\ $21$\,cm power spectra from the CD/EoR.
