I did hear from GitHub support:
This error is often caused by your network blocking the attempt by Windows to check the revocation status of a certificate, which then causes the whole operation to error. As this is affecting two Windows machines, with differing recent updates, it does sound likely that this is a network problem.
You can find a workaround for this issue in our GitHub Desktop repository here:
Can you give that a try and let me know how you get on?
I followed these instructions carefully, and I still have the problem. It shows that there is a very serious problem with GitHub on Windows that started a day or so ago. Evidently it is NOT my particular problem but a system-wide problem. It’s tempting to think it’s the result of the massive Windows 10 update but 1) my Windows 10 laptop showed the problem BEFORE it got the update and 2) some of the reports here involve older versions of Windows. I conclude that something very serious is wrong with GitHub.