Hi all, thanks for the replies.
As at least a couple of you suspected, my x1650 is apparently responsible. I've noticed quite a few quirks related to this card since I got it last year. I looked through the debug logs and noticed that GLSL was no longer compiling when I ran FS2open, so I tried downloading the atioglxx.dll file recommended in the troubleshooting FAQ.
Immediately, all the bells and whistles of FS2open worked beautifully and the input delay essentially disappeared. It's still barely noticeable when using the keyboard controls, but it's so small that it may well be either my imagination or the normal game mechanics.
Thanks for the feedback, everyone! I'm off to hunt some shivans!
(who am I kidding... they're the ones hunting me...)