How do particles in merging black holes "know" they need to change direction in time towards a new singularity? I have read this question:

Therefore there is nothing (except things travelling towards the singularity) inside the event horizon of a black hole.

Does anything exist between singularity and the event horizon of a black hole
Now imagine particles moving towards the singularity inside two black holes, that are actually merging. Until the horizons do not overlap, all particles are moving towards the singularity of their own black hole, that is understandable.
Now let's see what happens when the horizons start overlapping. How do particles that are already on the way towards their own singularity, "know" that they need to change course (either towards the other singularity or to a new common one)?
Here is what I do not understand: information inside a black hole can only travel towards the singularity (decreasing r), there is no way for information to travel backwards. Now when the black holes' event horizons start overlapping, naively thinking, the "travel towards the singularity" has to change direction, whatever that direction means, because on this site it is often mentioned that moving towards the singularity is like moving towards tomorrow (so travel in time not space).
Now information can only travel towards a decreasing r, that is equivalently, changes in the gravitational field can only propagate at the speed of light towards a decreasing r. But if that is true, then the particles that are moving towards their own singularity would have to receive information from other regions inside the new common black hole, so that information would need to travel outwards, that is increasing r. But that is not possible, and changes in the gravitational field cannot propagate outwards (increasing r).
So there are two things coming to mind:

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*particles moving towards their own singularity would need to make a change in direction moving in time as soon as the event horizons start overlapping


*to do that, particles would need to receive information (changes in the gravitational field) that is propagating outwards (increasing r) in the new common black hole
Question:

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*How do particles in merging black holes "know" they need to change direction in time towards a new singularity?

 A: In General Relativity, gravitation and the structure of spacetime are one and the same. When the two black holes merge, the gravitational field is already encoding the information of "where is the future", and in fact this is precisely why things fall.
You are correct that the "wiggles", so to speak, in the gravitational field take some time to be carried around. For example, when two black holes merge, the information of the merger is carried at the speed of light before it reaches the gravitational wave detectors at Earth. However, if a particle is in a region where the two black holes are touching, then by construction it is in a region of spacetime where this information is already present. Both black holes are distorting spacetime in that direction and their gravity is defining where the future points to.
Intuitively, we can think that the direction of time changed due to the gravitational influence of the new black hole that approached the original system comprised of particle and black hole. As the new black hole comes nearby, it warps spacetime differently and changes the way the particle will move. In terms of the (correct) statement that falling down the singularity is going to the future, the new black hole warps spacetime as to change "where the future is". At sufficiently late times, the black holes will have merged completely and set down to an stationary state, when a single singularity will be present.
Two side comments are in order to avoid misunderstandings:

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*I should point out that once the two black holes touch we can no longer distinguish them, since the event horizon possesses no local effect. In other words, one can't tell where one black hole starts and the other one ends. To think of two touching black holes as an analogy to get a better understanding of the process can be pedagogical, but it shouldn't be taken too seriously.


*As mentioned by safesphere on the comments, we must also recall that an external observer never sees anything falling down the black hole. This includes the star itself. Hence, an external observer will never really see the black hole forming, much less see two singularities merging. As a consequence, an external observer will see the two black holes merging before they see any singularity forming, so they don't really see any singularities merging or anything of the sort. If the observer falls down the black hole, when they finally see a singularity there will already be a single one.
Notice that, in any case, the problem of "how does a particle knows the direction of time?" does not depend on spacetime having singularities at all, and the very same problem could be posed on other spacetimes. For example, the very same question would hold if one were to ask "If a star enters the solar system and passes close to the Sun, how would the Earth 'know' where it should fall to, given it was already falling towards the Sun?".
For a bit more on "an external observer never sees a black hole forming", see, e.g., this answer.
