  Hyperuniform point patterns can be classified by the hyperuniformity scaling
exponent $\alpha > 0$, that characterizes the power-law scaling behavior of the
structure factor $S(\mathbf{k})$ as a function of wavenumber
$k\equiv|\mathbf{k}|$ in the vicinity of the origin, e.g.,
$S(\mathbf{k})\sim|\mathbf{k}|^{\alpha}$ in cases where $S(\mathbf{k})$ varies
continuously with $k$ as $k\rightarrow0$. In this paper, we show that the
spreadability is an effective method for determining $\alpha$ for quasiperiodic
systems where $S(\mathbf{k})$ is discontinuous and consists of a dense set of
Bragg peaks. We first transform quasiperiodic and limit-periodic point patterns
into two-phase media by mapping them onto packings of identical nonoverlapping
disks, where space interior to the disks represents one phase and the space in
exterior to them represents the second phase. We then compute the spectral
density of the packings, and finally compute and fit the long-time behavior of
their excess spreadabilities. Specifically, we show that the excess
spreadability can be used to accurately extract $\alpha$ for the 1D
limit-periodic period doubling chain and the 1D quasicrystalline Fibonacci
chain to within $0.02\%$ of the analytically known exact results. Moreover, we
obtain a value of $\alpha = 5.97\pm0.06$ for the 2D Penrose tiling, which had
not been computed previously. We also show that one can truncate the small-$k$
region of the scattering information used to compute the spreadability and
still obtain an accurate value of $\alpha$. The methods described here offer a
simple way to characterize the large-scale translational order present in
quasicrystalline and limit-periodic media in any space dimension that are
self-similar. Moreover, the scattering information extracted from these
two-phase media encoded in the spectral density can be used to estimate their
physical properties. (abridged)
