I wouldn't even assume that mathematics undergrads understand manifolds. I think the majority our math majors never see the definition of one before graduating, and only a small handful could actually tell you the definition (manifold is actually a really tricky notion!). In fact, one can even finish our "graduate prepatory track" without encountering the notion. We do have a differential geometry elective which probably contains the definition, but is focused on submanifolds of Euclidean space. In the US, I think outside of a few top places, it's not regarded as a standard part of the undergraduate curriculum; manifolds and their basic properties are the first thing covered in our graduate topology class, and I don't think we assume that students have any experience with them.

Very serious theoretical physics students may actually be more exposed to manifolds, since they take general relativity, but I doubt that actually includes much interesting exposure to the notion (since generally, they'll work in Euclidean space).

I think it's unlikely that any other large population of undergraduates is getting exposed to them; I certainly have never heard anything that suggests so, though of course, it's hard to rule it out happening somewhere.

how much exposition is necessary for it to be readable by scientists, engineers and related professionals: Basically, all of it. But do you really want to write a textbook on manifolds into your dissertation? If it were me, I would probably just start with "Let $M$ be a manifold. (For an introduction to manifolds, see the textbooks [3], [7], [34].)" $\endgroup$ – Nate Eldredge Jan 14 '17 at 4:57