On the implementation of the spherical collapse model in cosmology Lecture notes on the spherical collapse model found online (https://www.uio.no/studier/emner/matnat/astro/AST4320/h12/undervisningsmateriale/spherecollapse.pdf is one of them) consider a spherical top hat perturbation of some finite size, and argue via spherical symmetry that only the mass inside the shell under consideration is relevant to the dynamics of the spherical perturbation.
The value of the density contrast (relative to an Einstein deSitter universe) dictates the dynamics of the scale factor within that region: an overdense region is akin to a particle in a central force field with a velocity lesser than an analogous escape velocity, and so it has a cycloid trajectory in time. Similarly, an underdense region will have a hyperbolic trajectory that 'escapes out to infinity' and retains a finite velocity over there. On the boundary lies the critical density (corresponding to the EdS density), that moves at escape velocity.
Considering the overdense case, the trajectory in time is given by:
$\begin{equation}
a = A(1 - cos(\theta))
\\
t = B(\theta - sin(\theta)),
\end{equation}$
where $A$ and $B$ are coefficients that depend on the value of the density contrast inside the spherical region.
My concern with this model is that the scale factor $a$ at time $t = 0$ has a value of zero; meaning that independent of the comoving radius (say, $\chi$) of the region, the region starts out at a physical radius of zero, reaches a maximum radius, and then recollapses back to either a virialization or the origin itself. This then does not seem to model a perturbation that starts out at a finite physical radius or volume.
I'm aware that the scale factor in this case corresponds to a measure of separation between two neighbouring points, and so what the solution above really says is that the radial separation between two very close points that starts out at nearly zero has a cycloid trajectory in time. However, the same scale factor is used in computing the volume of the region as well, and so I wasn't able to resolve this 'volume going to zero' discrepancy.
Put in a different way, how does one incorporate an 'initial radius' condition while computing the volume of the perturbation in time?
 A: Let's replace $a$ in your expressions by $r$, so that we can discuss the global expansion factor $a$ separately. So we have
$$
r=A(1-\cos\theta),
\\
t=B(\theta-\sin\theta).
$$
Under the approximation of spherical symmetry, these equations describe the evolution of a region of physical radius $r$ that encloses density greater than the critical density. Separately, during our universe's matter-dominated phase, the scale factor evolves as
$$
a = \left(\frac{3}{2} H_0\sqrt{\Omega_M}\, t\right)^{2/3},
$$
where $H_0$ is the Hubble constant and $\Omega_M$ is the matter density today in units of the critical density. I believe that you are essentially asking what the initial comoving radius $\chi\equiv r/a$ is. Is this correct?
At early times in the spherical collapse solution, $\theta\ll 1$, so a small-angle expansion yields up to order $\theta^3$ yields $r=A\theta^2/2$ and $t=B\theta^3/6$ and hence
$$
r=A\left(\frac{9}{2B^2}\right)^{1/3}t^{2/3}.
$$
Then we can divide the above two equations to show that at early times,
$$
\chi_i\equiv \left.\frac{r}{a}\right|_\mathrm{initial} = 2^{1/3}A\left(B H_0\sqrt{\Omega_M}\right)^{-2/3}.
$$

That may suffice to answer the question, but we can go farther. If we expand up to $\theta^5$, we get $r=A(\theta^2/2-\theta^4/24)$ and $t=B(\theta^3/6-\theta^5/120)$, which leads (after some work, e.g. equations 34-42 of the manuscript that you linked) to
$$
\chi = \chi_i\left(1-\frac{a}{2^{2/3}5 B^{2/3}H_0^{2/3}\Omega_M^{1/3}}\right),
$$
where we are describing how the comoving radius $\chi$ evolves with the scale factor $a$ at early times. The density contrast $\delta\equiv \rho/\bar\rho-1=\chi_i^3/\chi^3-1$ is then
$$
\delta
=
\left(1-\frac{a}{2^{2/3}5 B^{2/3}H_0^{2/3}\Omega_M^{1/3}}\right)^{-3}-1
\simeq
\frac{3a}{2^{2/3}5 B^{2/3}H_0^{2/3}\Omega_M^{1/3}}
$$
at early times, which reproduces the well known $\delta\propto a$ from linear perturbation theory. This means that if we have an initial perturbation of comoving radius $\chi_i$ and initial density contrast $(\delta/a)_i$, then we can invert the above equations for $\chi_i$ and $\delta$ to get
$$
A=\frac{3\chi_i}{10(\delta/a)_i},\\
B=\frac{(3/5)^{3/2}}{2H_0\sqrt{\Omega_M}(\delta/a)_i}.
$$
Together with the original spherical collapse equations, this describes how an region of initial comoving radius $\chi_i$, whose density contrast is $\delta=(\delta/a)_i\, a$ at early times, evolves in the nonlinear regime (where $\delta$ is no longer much smaller than 1).
