As I mentioned the other day, disaster ensued when I hit the "update to Wordpress 4.7" button on one of my sites. Hostgator (not a referral/commission link, I am just a customer who loves them) was able to get me out of that mess by turning off an incompatible plug-in.
Well, we thought they got me out of that mess by turning off the plug-in. The site seemed to work fine from my end. I could put new content in and, when logged in, the site looked fine. But visitors who weren't logged in didn't see updates for three days and some of them (thanks, Mama! Thanks Aria! Thanks, Timothy) wrote to let me know about it.
I futzed around with various possibilities and finally tuned back to Hostgator. It took them a few minutes to find the problem -- something wrong with the .htaccess file, presumably created by the Wordpress update fiasco. Deleted .htaccess (Wordpress re-creates it if it's not there, although if you have customizations you will want to hand edit instead of delete) and things went back to normal.
So, if you had the initial update problem and are now having a "site doesn't update for not-logged-in visitors" problem, try that.