I've got it figured out now.
The old videos are fine; they play just fine, no changes required. The videos affected are those we've posted since the changes at YouTube, so perhaps the last three days.
The problem is a back-end issue, ie technical. WordPress has a unique way of posting YouTube videos, and the new YouTube format doesn't work with that. Between WordPress and YouTube, they need to work this out somehow. Ordinarily I'd drop a feedback to staff, but of course it's the weekend and they are closed. I suggest that bright and early Monday morning everybody who has had this issue drop a feedback.