I know that experiments like electron scattering can give the nuclear charge radius and proton scattering can give the nuclear matter radius. However, these experiments seem to first assume that the nucleus is spherical so as to calculate its radius. How do we know that the nucleus is spherical in the first place?
We have a variety of methods for detecting the fact that many nuclei are not spherical, and for measuring their deformations.
Classically a rotating rigid object has an energy given by $L^2/2I$, where $L$ is the angular momentum and $I$ is the moment of inertia. We also find that many nuclei have bands of energy levels with energies that go approximately like $L^2$ (or, usually more accurately, $L(L+1)$). Quantum mechanically, a sphere can't undergo this sort of collective rotation, because quantum mechanics can only describe motion as a change in the state of a system, and a sphere doesn't change when you rotate it. Therefore, when these bands are observed, the interpretation is that the nucleus cannot be spherical.
If you want to measure the deformation quantitatively, there is a variety of techniques. For nuclei that you can gather in macroscopic quantities, here are some techniques that can be used:
Coulomb excitation measures the transition quadrupole moment from the ground state to excited states. (And sometimes multiple Coulomb excitation is possible, so you get access to excited states as well.)
The nuclear hyperfine splitting depends on the static quadrupole moment of the ground state, if the spin is $\ge 1$.
Electron scattering can give information about deformation.
Most nuclides can't be produced in bulk quantities, so in most cases, a much easier technique is to measure the lifetimes of the excited states in a rotational band. If you think of the nucleus semiclassically as a rotating electrically charged ellipsoid, then the rate of radiation is proportional to the square of the quadrupole moment. For nuclei created in-beam in an accelerator experiment and studied through gamma-ray spectroscopy, the lifetime can often be inferred from the statistical distribution of the Doppler shifts of the recoiling nuclei, which are recoiling when they are formed by fusion and then slow down inside the target.
Sometimes one might know only the ground-state spin and parity. In many cases, this can be compared with theory to give a good idea of the deformation.
For a nucleus that is deformed, you might think that you could infer the shape by looking at the energy as a function of $L^2$ and getting the moment of inertia from the slope of the graph. This is a great method for molecules, but for nuclei it doesn't work very well, because there is a lot of model-dependence in the moment of inertia. Nuclei act like superfluids, so their moment of inertia is much smaller than the rigid-body value.
Most nuclei are either spheres or prolate ellipsoids, or unstable shapes that vibrate back and forth between such shapes. There are basically no known cases of stably oblate nuclei. Most prolate deformations are about as much as a chicken egg, but in some cases, e.g., fission isomers, the ratio of the lengths of the axes can be as much as 2 to 1. There is some evidence for pear shapes, but this is rare and most such cases are actually unstable shapes that vibrate between sphere and pear.
The measurements of quadrupole moments of nuclei can give an idea of the charge distribution of nuclei. It it generally assumed that the charge distribution corresponds well with the mass distribution. Very small quadrupole moments generally indicate spherical and near-spherical ground state shapes.
Quadrupole moments for excited energy states in nuclei can indicate the static charge distribution. Small moments along with moderately large first-excited energy levels ((> 800 keV) and equally-spaced higher levels generally indicate a vibration of the nucleus away from a basic spherical shape or the attachment of a single charged particle to a spherical core.
Large quadrupole moments along with low first-excited state energies (<300 keV), along with a gradually increasing spacing for higher energy levels generally indicate a rotation of a non-spherical charge distribution. The basic non-spherical static shapes are prolate (rugby ball shape) spinning about an axis $\perp$ to symmetry axis, and oblate ( circular cake shape) spinning about its symmetry axis. In motion, the prolate shape gives the effect of a smeared-out oblate shape, but the quadrupole moment is usually distinctly different.
An angular distribution of coulomb and coulomb-nuclear excitation scattering of alpha particles from nuclei can be used to extract quadrupole (and hexadecapole) moments. These can be used to infer static nuclear shapes.
Angular distribution of gamma rays in a 2nd-excited$\to$1st-excited$\to$ground state cascade can also be used to determine moments.
The data from scattering experiments fitted with form factors give the shapes.
The probability amplitude for a point-like scatterer is modified by a form factor, which takes into account the spatial extent and shape of the target.
These give the shapes of the nuclei.
The first experiments were electron scattering experiments on various nuclei.
The shapes are not just spherical, also discus shaped or other symmetric shapes, and recently even pear shaped (which leads to possible CP violation proposals).