In this question OP asks for a prediction for the expansion or contraction, therefore a scale parameter $a(t)$, when Universe is not neutral but uniformly charged.
A result for this would be
In a homogeneous universe filled with charged dust the Faraday tensor must be zero everywhere by symmetry (in spite of the charge) making the Einstein-Maxwell equation reduce to the standard Einstein field equation. Electromagnetism plays no role in determining the spacetime geometry.
As DFJ states
However, intuitively I still think it is strange that a uniformly charged universe is the same as one with no charge.
I also find it odd. I'd like to ask you if (Question:) you are in knowledge of any models for a Universe that's "slightly" charged. With this I mean that: we do expect the Universe to be neutral because we see gravity as the force describing its dynamics despite Coulomb's law being $\sim39$ orders of magnitude stronger, but at the same time if positive or negative charges exceeds by only a small amount, this effect might not be noticeable when studying the motion of objects.
This might be equivalent to ask for a model of Universe being perturbed by a small amount of charge. How would it be?
