  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]
