On a whim, I opened up the Downloads list (accessible by pressing Ctrl-J). It was pretty lengthy; I hadn't cleared it in as long as I could remember. (Admittedly, I just never bothered, as I didn't think it necessary.)

Also on a whim, I clicked Clear List, which wiped the slate clean, so to speak. (This doesn't delete your downlods, but merely Firefox's record of them.)

And you know what? The next time I downloaded a file, Firefox hummed along like its regular speedy self. No hangs, no delays. I'm not sure why this made the difference, but it definitely did. If you're encountering a similar issue, I recommend trying the same procedure.

(Then let me know if it helped, as I'm not sure if this was isolated to my PC or what.)