The video linked in this answer shows how space-time would be distorted during such an event. No doubt such an event would be an extremely strong source of gravitational waves. But would anything be visible in the electromagnetic spectrum (providing there is no other matter around)? Standard matter falling into a black hole strongly radiates, being torn by tidal forces. But what would happen to a black hole being "spaghettified" like this? Could its event horizon be somehow torn? Or would the event horizons just merge? Would there be any significant Hawking radiation emitted?
First, the video which the question links to augments real data with artistic interpretation. I am sure that the OP and other posters know that, but I just wanted to make sure there was no misunderstanding.
I was part of the research group that created the data shown in the video. We worked with a scientific visualization specialist/artist to create the video. He added the background of stars.
The short answer to the question is that there would not be any electromagnetic (EM) radiation beyond the EM radiation already produced by charged matter falling into the black holes.
Putting aside the issue of unifying general relativity and quantum mechanics, the event horizon is meaningful only for an isolated black hole. Typically, this is a Schwarzchild (non-spinning) or Kerr (spinning) black hole, though it could be more exotic. Isolated means that the black hole exists in a universe devoid of all other matter and energy. In other words, a black hole that exists on paper only. In such a case, the event horizon is that the boundary between light paths that never move away from the black hole and those that do. Finding that boundary requires an exact solution to Einstein's equations, and we only have solutions for a few special cases, which do not include astrophysical (real) black holes.
To understand the collision (or merger) of two astrophyical black holes, we either find approximate solutions or solve the equations numerically on computers (simulate). In this case, we consider at apparent horizons. These are boundaries between light paths that move away from the system of black holes and those that don't. Defining this precisely requires a choice of coordinate systems, which is a rather involved topic that probably goes outside the OP's interest. Nonetheless, the apparent horizon is probably what the OP is asking about. It corresponds best to the intuitive idea of the event horizon.
I simulated mergers of two black holes, and in the simulations, the apparent horizons of the two black holes joined into a new apparent horizon that looked initially like the shell of a peanut with two lobes. The details of the formation depended on the coordinates chosen, so there isn't a single sequence of apparent horizons to consider.
Apparent horizons and event horizons are not physical boundaries. They are divisions of spacetime. An analogy is the distance at which a jet fighter no longer has enough fuel to return to the aircraft carrier from which it took off. Nothing happens at the point, and there is nothing to see, but after the plane exceeds that distance, it cannot return to the carrier. So the apparent horizon of merging black holes is a solution to a partitioning of spacetime according to light paths, not a physical surface.
The details of the matter falling into the two blacks holes would be sufficiently complicated, that the details of the apparent horizons before the merger and the merged apparent horizon after the merger would be insignificant. The matter produces EM radiation only if it is charged, and the charge on the matter would be caused by the high energy collisions of the matter particles as they are accelerated toward the black holes. That is an incredibly complicated phenomenon.
The spaghettification occurs within the apparent horizons of the black holes. For it to occur, there must be enough difference in the force of gravity between the closest and farthest points on the body that is falling into the black hole. By that point, no light produced in that process can leave the black hole, so we can't see it.
Finally, though the merger of the black holes would not produce additional EM radiation, we could observe the EM radiation of the black holes as they spiral in to merge. That is similar to the observations of PSR B1913+16, a binary neutron star where one of the neutron stars is a pulsar radiating EM waves. This system provides evidence that supports general relativity. Also, we could see the EM radiation of the black hole formed by the merger, which could have a complicated variation as the black hole settles into a steady state.
Not sure about overall EM radiation from colliding singularities, but one part is almost possible to answer: the Hawking Radiation (HR). As singularities approached, the amount of radiation emitted would reduce slightly in certain directions. HR emitted by Black Hole and 2 (S1 and S2) should be absorbed by the other. Assuming S1 and S2 then have a significantly greater mass, the amount of HR emitted after the collision would also be significantly reduced.
This is all assuming the (very unlikely) situation of no massive amounts of angular momentum being involved between S1 and S2.
Theories about Black Holes with large angular momentums vary widely, but there is a large amount of traction to suggest that the formation of Pulsars can happen around rapidly spinning singularities, and can create interesting shaped event horizons (look up the work of Roy Kerr). Some suggest that this can lead to the formation of "Naked Singularities" at which point all bets are off. This is far outside the world of what I'm comfortable about answering, but hopefully you have enough information to conduct a little more of your own reading.