In this lecture by Fredric Schuller it is said that in the case of a non compact four dimensional manifold there is a non countable infinity of differentiable or smooth manifolds that are NOT diffeomorphic.
Differentiable structures definition and classification - Lec 07 - Frederic Schuller
My question is that how this fact from math can be related to or affect the study of black holes, say finding the Schwarzschild solution or the study of cosmology, say solving for the FLRW metric.
I mean in which part of the calculations we specify which specific smooth structure, i.e. $C^{\infty}$-compatible maximal atlas are we using to take the chart from it and put a coordinate system?
There are two possible answers to this.
The prescriptive picture When talking about a particular solution as described in a particular coordinate system, your coordinate system is the smooth structure: in asserts that (a portion of) your manifold is diffeomorphic to (some subset) of a standard 4-manifold.
The dynamic picture For actually studying cosmology, say, or black hole evolutions, one typically obeys causality and only cares about globally hyperbolic space-times, which are known/constructed to be diffeomorphic to $I \times \Sigma$ where $I$ is an interval (of time) and $\Sigma$ is a three-manifold (of space). Three manifolds admit unique smooth structures, and so there's no issue with multiple compatible smooth structures on the space-time in this picture.