Upcoming weak lensing surveys will probe large fractions of the sky with unprecedented accuracy. To infer cosmological constraints, a large ensemble of survey simulations are required to accurately model cosmological observables and their covariances. We develop a parallelized multi-lens-plane pipeline called UFalcon, designed to generate full-sky weak lensing maps from lightcones within a minimal runtime. It makes use of L-PICOLA, an approximate numerical code, which provides a fast and accurate alternative to cosmological $N$-Body simulations. The UFalcon maps are constructed by nesting 2 simulations covering a redshift-range from $z=0.1$ to $1.5$ without replicating the simulation volume. We compute the convergence and projected overdensity maps for L-PICOLA in the lightcone or snapshot mode. The generation of such a map, including the L-PICOLA simulation, takes about 3 hours walltime on 220 cores. We use the maps to calculate the spherical harmonic power spectra, which we compare to theoretical predictions and to UFalcon results generated using the full $N$-Body code GADGET-2. We then compute the covariance matrix of the full-sky spherical harmonic power spectra using 150 UFalcon maps based on L-PICOLA in lightcone mode. We consider the PDF, the higher-order moments and the variance of the smoothed field variance to quantify the accuracy of the covariance matrix, which we find to be a few percent for scales $ell sim 10^2$ to $10^3$. We test the impact of this level of accuracy on cosmological constraints using an optimistic survey configuration, and find that the final results are robust to this level of uncertainty. The speed and accuracy of our developed pipeline provides a basis to also include further important features such as masking, varying noise and will allow us to compute covariance matrices for models beyond $Lambda$CDM. [abridged]
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