An observer falling into the black hole would not observe any distortion in time. They would simply fall in, under the influence of gravity. From the perspective of a far-away observer it would look as if time is slowing down as the photons would take increasingly longer to escape. At the event horizon the photons would effectively be held in place. Eventually though, the last photon will have escaped and you will just observe a slightly larger black hole.

So the merger definitely happens from the point of view of the black holes. We might observe odd artifacts but they would eventually fade away.