Thanks! That explains it! I’m a bit slow in replying as I did some further investigations. I saw that several people had this problem caused by Safari not using the native media player. It seems that this could be controlled in iOS 12 via Safari’s Advanced Experimental settings but not in iOS 12.4.1
I wonder if it the option will reappear in iOS 13? Or in padOS?
I tried several suggested tricks to to force Safari to use the native player but none worked for me.
I did find that if I opened the website in Chrome then the AirPlay icon did appear on screen when playing a video. So that fixed the problem!
I was trying to watch videos on WTATV.com which is a UK-only subscription-only website so you couldn’t have emulated me. But I note that I also can’t AirPlay videos from wtatennis.com/videos in Safari.
It seems odd that Apple don’t allow the use of their own media player but seem to remember that Windows did something similar with playing DVDs.