An update on this.
I bought a Chromecast, which has solved the Eurosport & Youtube problem. However, UKTV Play doesn't support Chromecast, although it looks as though it may at some time in the future - we need it as we're in the process of catching up with Dave Gorman's "Modern Life is Goodish" (highly recommended!) - so at the moment we need the Fire Stick as well.
Things are now getting interesting! As I suspected Google's withdrawal of Youtube from Amazon Fire was in retaliation for Amazon's refusal to sell Chromecast. This morning I got an email from Amazon saying that Firefox & Silk browsers are now available on Fire TV. Yeah, you guessed it!
The Fire TV version of Firefox is very basic, with hardly any of the features you'd expect of a browser & not supported by Eurosport Player, but it serves Amazon's purpose.
What will Google's next move be? One thing's for sure, the big losers in this silly quarrel are the users.
Having used both Fire Stick & Chromecast I think I prefer the latter - it's easier to go straight in from an app & it doesn't want to sell me Amazon Prime. . .