What happens to the electron if the nuclues instantaneously disappears? Imagine a hydrogen atom alone within an empty universe. Now imagine that the proton that the electron in the our hypothetical atom is bound to instantaneously disappears. From a quantum perspective, what exactly happens to that electron? Does its probability density ripple out into the surrounding universe?
 A: 
Now imagine that the proton that the electron in the our hypothetical atom is bound to instantaneously [my emph.] disappears. 

The thorny word here is instantaneously:  nothing can truly happen instantaneously (that is: event duration $\Delta t=0$) as it would violate Relativity's universal speed limit, $c$. So it's not a very interesting question because it's a purely hypothetical event.
Consider the time dependent Schrödinger equation (TDSE) for a hydrogen atom with a vanishing nucleus:
$$i\hbar\frac{\partial}{\partial t}\Psi(r,t)=\Big[-\frac{\hbar^2}{2\mu}+U(r,t)\Big]\Psi(r,t)\tag{1}$$
Where:
$$U(r,t)=-\frac{Z(t)^2e^2}{4\pi \epsilon_0r}\tag{2}$$
For a normal hydrogen atom $Z(t)=1$ and so the potential function $(2)$ becomes:
$$U(r)=-\frac{e^2}{4\pi \epsilon_0r}$$
With $U$ no longer time dependent, $(1)$ can then be solved the usual way and provides wave function solutions as per usual.
But in the case of a vanishing nucleus we'd have $Z(0)=1$ and $Z(+\infty)\to 0$.
The trouble is that when $U$ is time dependent, the usual and enormously simplifying 'trick' of separating the wave function into a time dependent part $T(t)$ and a spatially dependent part $\psi(r)$:
$$\Psi(r,t)=\psi(r)T(t)$$
(The separation method for the TDSE can be found here.)
... is no longer possible because $(1)$ is not a linear partial differential equation anymore.
If $Z(t)$ was known then possibly numerical methods could be used to find the electron wave function from $(1)$. $\Psi(r,t)$ would then describe the transition from bound to free particle. When the binding electrostatic attraction between nucleus and electron vanishes completely (when $Z(+\infty)=0$), the electron would be a free electron.
