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In this article, we study the weak gravitational lensing in the background of regular, static, spherically symmetric black hole solutions of Einstein's standard general relativity coupled with nonlinear electrodynamics. The weak deflection angles are estimated in the context of vacuum medium and plasma medium using the Gauss-Bonnet method. The obtained deflection angles decrease as the impact parameter $b$ and the charge parameter $q$ increase, while the deflection angles increase gradually with increasing values of the black hole mass $m$. Moreover, the effect of a plasma medium has increased the deflection angle than the vacuum medium scenario. We also estimate the time delay in the field of the described black holes that vanishes for $m = q =0$, i.e., the absence of the black holes. The shadow cast of the present black holes is also analyzed with respect to the impact $q$ and mass $m$, which ensures that the shadow region shrinks for increasing values of $q$ and expands for increasing values of $m$. In addition, we estimated the rigorous bounds of the greybody factor $\mathcal{T}_b$ for the described black holes and the graphical analysis ensures that the increasing charge parameter $q$ decreases the rigorous bound of $\mathcal{T}_b$ and the increasing mass $m$ increases the rigorous bound of $\mathcal{T}_b$.