HTTP is very useful for caching and mirrors for Linux distributions because signatures are checked by package managers outside of the transport protocol.
I suppose in your case the window would be smaller using HTTPS, as the package downloaded could probably be inferred only after the download is complete. But your method would need automation anyway, so I don't think the difference in timing would really matter in practice.
But if you are in the position to automatically attack someone, why wait to see if they are downloading a fix? Why not just try to attack with known recent vulnerabilities anyway?
That said, Linux distributions don't usually use browsers to pull packages, so the point is moot.