First, the many-worlds interpretation of quantum physics was originally mooted by Everett as a solution to the physics-philosophical question posed by the collapse of the wave function. In the popular literature, this is often posed in terms of the consciousness of the observer (in the Copenhagen picture). This is possibly due to the Kantian influence in Western philosophy when as Kant was interested at where human consciousness meets the world, but his question was of a different order and more profound than that of quantum physics as he was tackling Humes critique of causality - which is one of the conditions of science.
Now, one doesn't need a many-world picture to capture how the wave function collapses. For example, in the modern interpretation of the Copenhagen picture - the consistent histories or decoherence picture, consciousness is depromoted, one electron for example can 'observe' another. That other electron for example is external to the other.
One should be clear about what Everett is attempting to do. He is taking a theory of physics as a complete picture of reality and attempting to find a new prediction. This is a respectable procedure. After all, Dirac discovered the positron on physical grounds, Einstein discovered that matter & energy were inter-convertible. But notice that one can suppose that there was an electron whose charge is positive - it seems fairly straight-forward to imagine, that matter & energy are inter-convertible less so, but it at least carries on a tradition on physics where energy is seen as a dominant paradigm. But the idea of bifurcating realities at every point of space & time at every possible interaction of electron with a photon is of an order of speculation vaster than any previously attempted! This of course does not mean that it is wrong, but one should really pause for some serious reflection as to what kind of picture of reality it is giving, and how tenable it can be. Recall, that any theory of physics is always contingent. One needs to then understand what the predictions of any one theory is, given how we see reality over all - which is essentially a common-sensical, philosophical, or theological view.
I don't know enough about the philosophy of time to be able to give any sense of the contours of this question, but it does seem like a coherent question. The idea of time flowing equably at every point in space is a condition for Newtonian physics. It is generally known as absolute time. (There was also a notion of absolute space which had been abandoned by Newtons time as untenable). Liebniz had a very different picture of what space & time was which was relational rather than absolute. And there was a famous spat between Newtons supporters & Liebnizs, which to a large extent was driven by chauvinism than ideas (one still notices the political differences between the continent & the UK now).
One notices in Einsteins theory that absolute time is abandoned. Simultaneity becomes a more flexible notion. Every particle carries its own clock. Of course seeing that every particle moves in spacetime what can it mean for a particle to carry its own clock? It means that the clock measures time for them as they move.
Ishmael & Layla are together, their consciousness are at the same time & place because they occupy the same time & place, because that is what being together means. But Ishmael decides to move away, can one say that their consciousness is now together or simultaneous? From a distance Layla sees that Ishmaels consciousness has slowed down paradoxically because his life has accelerated. They both move on. He returns. He isn't much older, she has aged. Time has moved at a pace different for each of them. Can one say then that consciousness has been for both of them simultaneous for both whilst they were together and whilst they were apart and again together?
If we lived in a possible world where the speed of light was not much higher than walking speed we would have a profoundly different idea of what time is and how it operates as we would be immediately conscious of differences that can only be intellectualised now.