Very interesting ideas, everyone!

In regards to what happened to Narnia after the Pevensies disappeared... aside from the initial concern and shock over the kings and queens disappearing, it may not have been as difficult as you would expect. Archenland was still around, for one thing, so you still had their courts for settling disputes. Also, in the absence of the Pevensie siblings, the monarchy of Archenland had a fairly legitimate claim to the throne of Narnia. The founder of the country and its royal line was the second son of King Frank; they come from essentially the same bloodline that ruled Narnia before the effects of the Tree of Protection wore off and Jadis attacked Narnia. So, it would make sense to me that they might rule in proxy when the Pevensies disappeared.
(I'm guessing that the reason why the Archenlanders couldn't fulfill the prophecy and destroy the hundred year winter is because none of them are technically human. They're all a mix of human blood and dryad, naiad, river god, and wood god blood.)
Still, if Narnia had Archenlander leadership, I can't help but wonder why and how the country was defeated so quickly by the Telmarines when they invaded about a thousand years later. As I think about it, it seems possible that—after the course of so many years—most of the Archenlanders' blood had mixed so much with the previously mentioned nature spirits, they weren't really human
at all anymore.
If this was the case and if the Telmarines entered Narnia and began industrializing the country by the way of chopping down trees, damming waterways and building bridges, then I'd say it's logical enough to think that a lot of what was left of the barely-human Archenland bloodline would be trapped, killed, or at least frightened into hiding with everyone else. It's interesting, too, because the trees and waters waking up and taking Narnia back is a major theme in
Prince Caspian, so it seems like it could fit well. For all we know, the river god that ended the Second Battle of Beruna could be King Frank and Queen Helen's great-x100-grandson.

Kind of a weird theory, but I'd say it's relatively plausible, too. And sorry, I didn't mean to get so off-topic with this post.
To bring it back full circle... I definitely agree with everyone else; I think it would have been really interesting if Lewis had written a story about this era of Narnian history. Aside from the little bit of information we see on the Timeline that tells us that (the second?) Queen Swanwhite lived in the middle of it, we don't know very much about what happened during those thousand years at all.