I'm trying to make a cosmology, and I would like to know if my reasoning for the cosmology is correct and how large it would be considered.

So our universe has like three infinite continuous spatial dimensions. By adding another spatial dimension, there's enough room to fit an uncountable infinite number of alternate universes all lined up (think of an infinite lineup of sheets of paper, now scale everything 1 dimension higher). However, then someone in one universe can't traverse to another without crossing all the universes in between. This makes things crowded, so we need more room.

So we'll add another dimension, so I can open up a portal to another universe, and my friend can open up another portal to his universe, and no matter which we pick, we won't intersect any other universes.

In fact, because we also want each universe to be infinitely curled up so that we can achieve seemingly faster than light travel within each universe by moving in a different spatial dimension, we should add an extra dimension so that there's always a way to reach other universes nearby while also never having instances where an intersection with another universe has to happen.

If we want to throw extra universes with higher dimensions into the mix, that'll just make things more complicated. To make things simpler, we'll simply have an uncountably infinite number of dimensions. That way, whenever someone travels in one direction, there's an infinite number of other perpendicular directions we can go to that don't happen to reintersect with the current universe or other universes. It means that the entire universe or at least large portions of it are potentially within short distance as long as you know the correct direction to go, while keeping certain dimensions still a bit far away, and still needing potentially faster than light travel.

So how big is an uncountably infinite number of dimensions, each continuous? Does it change if the universe's size is an infinitely dimensioned sphere with a finite radius that spatially loops around at the edges?

The universe is equivalent to the set of all functions (not just continuous functions) from R->R.