Looks like I finally succeeded in getting the Linux Opera browser, fixed. I tried a few times to notify Opera of the problem of Flash video not playing on the bbc.com website.
I finally found a link to submit an Opera bug and about a day later, an update to the Opera browser came through the Mint 18.3 Update.
Now, the Flash video is once again working, on the bbc.com website. :-)
Cool! Btw, for anybody that uses the Chromium/Chrome browser, you will find the settings area looks almost exactly the same. Opera's browser engine has been based on Blink, the same engine that Chromium and Chrome uses, since 2013.