It's certainly possible to come up with equations of the trajectories in terms of the quantum numbers. We first find the general elliptical/circular paths in a classical fashion, and then see if the quantization rule gives us some constraints.
Let's work with polar coordinates in the plane of the ellipse, using coordinates $r$ and $\theta$, centered at the nucleus; we will move to three-dimensional spherical coordinates shortly. Considering the acceleration along $\mathbf{\hat{e}}_r$ using the EL equation, we have (taking $\mu$ as the reduced mass of the system; if the electron has mass $m_e$ and the proton has mass $m_p$, then $\mu=m_em_p/(m_e+m_p)$) $$\frac{e^2}{r^2}=\mu\ddot{r}-\mu r\dot{\theta}^2,$$ where we have used the inverse-square force by a nucleus with one proton.
Along $\hat{\mathbf{e}}_\theta$, we have $$\frac{\mathrm{d}}{\mathrm{d}t}\mu r^2\dot{\theta}=0.$$
I'm not going to go through the whole process of solving these equations, but it isn't too elaborate. We introduce a few variables which arise either for convenience or as constants of integration, and present a solution:
- $p$ is the total angular momentum of the system; $\mu r^2\dot{\theta}=p$.
- $W$ is the total energy of the system when you consider the center of mass to be at rest. We desire negative values of $W$, which are indicative of bound states here.
- $u=1/r$.
$$u(\theta)=\frac{e^2\mu}{p^2}+\frac{1}{2}\sqrt{\frac{4\mu^2e^4}{p^4}+\frac{8\mu W}{p^2}}\sin\theta$$
It turns out that this is the equation for an ellipse with a focus at the origin; the semiminor and semimajor axes $a$ and $b$ are given by $$a=-\frac{e^2}{2W};\quad b=\frac{p}{\sqrt{-2\mu W}}$$
We have not yet shown that only certain values of $a$ and $b$ are allowed, since there are presently no restrictions on $W$ and $p$. But if we can find the allowed energies and total momenta, it's trivial to make substitutions and arrive at the desired form of the description of the ellipse.
To observe quantization, we can apply the Wilson-Sommerfeld rule for each of the spherical coordinates $r$, $\vartheta$, and $\varphi$ (think about how this compares to the previous system of $r$ and $\theta$; it's relevant). We have
\begin{align}
\oint \mathrm{d}r\,p_r&=n_r h\tag{A}\\
\oint \mathrm{d}\vartheta\,p_\vartheta&=n_\vartheta h\tag{B}\\
\oint \mathrm{d}\varphi\,p_\varphi&=n_\varphi h\tag{C}
\end{align}
Equation C is the easiest. $p_\varphi$ is a constant, as described in the Rotator section of the page linked in the question; we introduce the magnetic quantum number $m$ to define the projection of angular momentum onto the $xy$ plane as $$p_\varphi=m\hbar;\quad m=\pm1,\pm2,\dots$$
There are a few ways to solve equation B; I find it easy to return to our old formalism with $\theta$ which we used while finding the orbits' shapes; it resembles the approach used in the paper linked by G. Smith in the comments (https://arxiv.org/abs/1605.08027). We know $p_\theta$ is the total angular momentum, so $$p_\theta=p=p_\vartheta+p_\varphi.$$
We introduce the azimuthal quantum number $\ell$; $$\oint\mathrm{d}\theta\,p_\theta=\ell h\Rightarrow p=\ell\hbar;\quad \ell=1,2,\dots$$
We have thus successfully enforced the quantization of $p$; we hope that equation A helps us resolve $W$. It's a relatively long process; we re-use our old definitions of $u$ and $\theta$, $$p_r\mathrm{d}r=\mu\dot{r}\mathrm{d}r=\frac{p}{u^2}\left(\frac{\mathrm{d}}{\mathrm{d}\theta}u\right)^2\mathrm{d}\theta$$
Using our expression for $u(\theta)$ and applying equation A, we have a very troublesome $$p\epsilon^2\int_0^{2\pi}\mathrm{d}\theta\,\frac{\cos^2\theta}{(1+\epsilon\sin\theta)^2}=n_rh,$$ where we have introduced $\epsilon=1-\frac{b^2}{a^2}$.
Fortunately, we are given a rather neat solution to this: $$p\left(\frac{1}{\sqrt{1-\epsilon^2}}-1\right)=n_r\hbar$$
We compare this to our formula for $p$ in terms of $\ell$ to find a relationship between the parameters $a$ and $b$ of the ellipse, and solve for $W$. This gives us $$W_{n_r,\ell}=-\frac{\mu e^4}{2\hbar^2(n_r+\ell)^2}.$$
You can plug these expressions for $W_{n_r,\ell}$ and $p_\ell$ into the expression for $u(\theta)$ to find the equations of the elliptical orbits you desired.
We may now analyze this result to answer the second part of the question. Clearly, the degeneracy of the orbits with the same $n_r$ and $\ell$ is implied by the allowed energies. There are a couple of other important visual notes regarding the shapes of orbits:
- We generally have elliptical orbits; they're circular when $n_r=0$.
- $m$ is indicative of the inclination of the orbit in space: the absolute value thereof dictates the angle between the plane of the orbit and the $xy$ plane, and the sign tells you whether the orbit is clockwise or anticlockwise.
Reference
Introduction to Quantum Mechanics by Pauling and Wilson
