Dimensionality has a very strict mathematical definition. Within that definition, a single universe with space and time has 3 space dimensions and one time dimension. The geometry of the universe means that moving objects have certain allowed trajectories and that the overall space-time has a certain geometry. From the definition of dimensions, time is one-dimensional in the sense that the time coordinate of an event (at least from one observer's point of view) is given by a single number. The spatial coordinates require three numbers, hence three dimensions.

Now, if the universes could branch, as in the MW interpretation, and you could somehow look down from above on all the branches, then you'd need additional dimensions to keep track of which universe you were in as well as the 3+1 dimensions within that universe. I'm not quite sure how this would work, though.

*However*, the fact is that you can't look down from above. The many worlds interpretation is, for any observer, indistinguishable from if there was only one dimension with the Copenhagen interpretation. So in every case, you can simply treat the universe as only having one time dimension (which is one-dimensional, I suppose). Questions about more dimensions due to multiple universes is essentially metaphysics, since none of it changes the 3+1 dimensional nature of the model we actually use to do calculations.

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Though I suppose theories like string theory, which propose 10-11 dimensions do indeed have extra dimensions within our own which can be accessed by observers. It's just that unless we look with a very powerful microscope (i.e. particle accelerator) we only see 3+1 of those dimensions. I really don't know what those have to say about time as a dimension.