I have another update for you:

After only a day, a Google search for "iOS 8 won't play video" has several more pages of relevant results than yesterday. The consensus of these results is that there does indeed seem to be a bug that prevents the display of HTML5 embedded video in iOS 8, but only when you've saved the mobile website to your home screen as a mobile web app. It seems (and please verify this if you would) that if you open mobile Safari and navigate to our mobile site at https://mobile.ballroomdancers.com/, the video will play there.

It's not ideal, but it's a decent workaround, and will do the job until Apple fixes the issue.

Regards,
Jonathan

Yes, it seems to be the case for now: we will have to use the mobile version through Safari. Not ideal but still better than searching the BDC full site on an iPhone screen. I will keep checking on the bug issue too. Please let me know if you have new solutions.

Thank you very much for the effort!