I’ll assume you understand that SketchUp does not have real circles: it represents a circle as a polygon with some attached metadata about the circle’s center, radius, etc. The metadata is created as you draw a circle or arc; there is no other way to generate it.
Because it can extrude along any path, follow-me always treats the path as a collection of edges, abandoning the metadata. So even if you “lathe” a shape by applying follow-me with a circular path, SketchUp won’t recognize the ends of the object to be circles.
There are really two parts to your question: how to find the center, and how to make the end be recognized as a circle. Since it is easier, I’ll take the second part first: just draw a new circle exactly on the end (be sure to use inferences carefully) and SketchUp will rediscover the circle. But that glib answer presupposes the solution to the first part, since you can’t draw the circle unless you can find the center of the end!
If there is something you can use to infer the center (e.g, the original circular path), then you really don’t need to do anything more - you can use the combination of that inference point and the inference on the end of the object to get the center point of the end. Notice in this animation how once I find the center of the circular path, SketchUp remembers it as a “from point” inference when I draw the circle on the top. You could likewise use that combination of inferences for any other purpose. Also notice that once I redraw the circular top, selecting an edge selects the whole thing. SketchUp created new metadata for the circle.
If there is nothing from which to infer the center, you can do as @colin suggested: draw diagonals across between opposite vertices of the polygon. They will intersect at the center of the original circle.