Plugins Not Updating. Setting Somewhere?

I just noticed that, suddenly, after moving our site to a VPS, all our plugins (and perhaps the core WP code) is no longer notifying me of updates. I also see that several of them are not up to date but the Updates link tells me everything IS up to date even though it’s clearly not.

Any idea where to look to fix this? I’m very concerned and afraid for our database which contains 8 years of posts.

Any help would be greatly appreciated.