Is the speed of light the same in a true vacuum and a false vacuum? Is there a difference in the speed of light in a true vacuum vs. a false vacuum?
 A: First, just to be clear, the speed of light is a misnomer in the context of relativity. We should distinguish between:

*

*The speed of massless particles (or, equivalently, the maximum speed of causal influences)

*The speed at which photons travel.

The reason I'm making this distinction is because a change in the vacuum state could lead to a change in the mass of the photon; however I take it this is not what you are interested in. Therefore, for the rest of this answer, I'll use "the speed of light" in the first sense.
Because the speed of light is a dimensionful variable, we can always choose our system of units so that it takes a given value (say, $1$). By performing a coordinate transformation to a locally inertial coordinate system, we can always arrange that light travels at $c$ in the neighborhood of any event.
The only physical quantities that are unambiguous and agreed upon by all observers are dimensionless quantities, like the fine structure constant. Dimensionless quantities absolutely can change as a result of false vacuum decay, if they are associated with background fields that change during the vacuum transition.$^\dagger$

$^\dagger$ Although, to be 100% clear, there is no experimental evidence suggesting that the value of the fine structure constant is associated with a field; it is a logical possibility given the rules of QFT but it's also logically possible that vacuum decay does not happen in our Universe or that the fine structure constant does not change.
