Consider two hypothetical versions of this. One, the exact scenario as you described - history unfolded like it did, until the 1900 alien incident. CS and information theory is in its infancy. You're correct that most of the necessary work would first go to physics and chemistry and their various spin-off fields, because that's what's needed to build tools necessary to inspect the machine in full detail. The math would develop along the way, and eventually enough CS to make sense of the observations made before.

Now for an alternate scenario: it's the 1900 again, with the twist that CS is already well-developed theoretical field of mathematics (IDK, perhaps the same aliens dropped us a mechanical computer in year 1800). We'd still need to push physics and chemistry (and spin-offs) forward, but this time, we would know what we're looking for. We'd know the thing does computation, we'd be able to model what kind of computation it does. The question would change from "what does this thing do" to "how exactly does it compute the specific things we know it does". I imagine this would speed up the process of getting a complete picture, because it's easier to understand a specific solution to a problem once you know the answer, than it is to figure out the answer along with the solution.

In terms of understanding the brain, we are in the second situation. We may still know little about how the gooey thing ticks, but we have a growing understanding of what comes out of all that ticking, and a very good understanding of the fundamental rules of ticking.