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:
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.