Technically/mathematically correct answer
Here's an example of a black hole that is technically inside another black hole: the maximally extended Reissner-Nordstrom solution. In this case, what we mean by being "inside" the black hole is that the event horizon for one black hole is completely to the future of the other (see below for why this isn't intuitively what you would think of as one black hole inside of another). This solution corresponds to a spherically symmetric spacetime with electric charge in the center of spherical symmetry. The Penrose diagram for this solution is shown here:
Starting at the horizontal dashed line and moving upward, we encounter the first black hole event horizon at the diagonal line labeled $r=r_+$. Passing this, we enter the first black hole, and unlike the Schwarzschild solution, there are now two separated timelike singularities at $r=0$. Because they are timelike, they can be avoided, so a timelike worldline can begin moving radially outward once it passes the first surface labeled $r=r_-$. Eventually this line will emerge out of a white hole horizon (at the next set of lines labeled $r=r_+$, and then will see there is another black hole horizon to the future. The maximally extended solution in fact consists of an infinite chain of black holes and white holes, and each is inside the next in the sense that each successive black hole is to the future of the previous one. Such a black hole chain also arises in the maximally extended Kerr black hole, describing a cylindrically symmetric spinning black hole in vacuum.
Of course, this spacetime is somewhat exotic. It possesses Cauchy horizons, for example at the first $r=r_-$ surface, so that initial data on the first slice does not completely determine how fields will evolve throughout the space time (you need extra boundary conditions at the singularity). Cauchy horizons are generically thought to be unstable, so it is unlikely such a black hole chain would ever arise in an astrophysical black hole. Perturbations to this spacetime tend to cause the chain to pinch off, leading to null or spacelike singularities.
You could probably cook up other examples of black holes with multiple asymptotic null infinity regions which has one black hole inside the other. It is likely that for such an exotic situation you end up violating an energy condition for the spacetime, or having other pathologies such as not being globally hyperbolic. In fact it might be possible to prove various theorems about global hyperbolicity or other aspects of the causal structure in these cases, although I don't know of any work examining this in detail.
More intutive explanation
I realized after writing the above answer that it doesn't really correspond with the intuitive notion of one black hole inside another.
The reason it is a little tricky to describe one black hole inside of another one is that black holes are defined in a global, "teleological" way, which means their definition depends on the entire history of the spacetime and not as we onften imagine them as being objects that exist at any given moment in time. When one thinks about one black hole being inside another, one would imagine a small event horizon inside the bigger one that eventually runs into the singularity. However, this is not possible because the small event horizon on the inside is actually not the boundary of the causal past of future null infinity. What this means is that there is not a good way to define the event horizon of this inner black hole, since the true event horizon actually only consists of the outer black hole.
If the smaller black hole originally started outside of the larger one, then the event horizon initially consists of two different pieces, but once they merge the event horizons combine into a single connected one, and so there is now only one black hole.
So then you asked what happens when the singularities collide. I think this is in general a fairly complicated question, since the equations of general relativity behave chaotically near singularities. Actually, it is fairly likely that not much is known about the structure of these merging black hole singularities, since numerical relativity is computationally expensive, and it becomes much harder when you are in regions of high curvature near the singularity. It think it is reasonable to guess that the full singularity is a single surface consisting of spacelike and null regions, but the details are complicated due to the chaotic behavior. When the singularities are very close, you would probably expect very rapid and violent contraction and expansion of space that is characteristic of these singularities.