Well…
I don’t know what to think about this.
The reply by @jackw.27 got me thinking it is all a network a problem; I had some UniFi firmware upgrades recently and an update to the Controller. But that didn’t make sense to me as it was only affecting HA and @NickRP7 described a gradual ‘degradation’ which I think I also experienced with my wall tablet going first.
So for some reason I decided to just try one more time before some taking some radical action. I used Incognito mode on Chrome on my tablet and Lo! it worked. Then I went back to Fully Kiosk, tried that and yes, it works now too!
I can categorically say nothing has changed since posting yesterday except for a few restarts of HA.
I know you won’t believe me 
So I’m happy it appears to be all working now but not happy that I don’t know why it is, or why it stopped working in the first place.
I’ll be keeping a nervous eye on things.
I do wonder if there is some underlying issue because I have discovered this is not the only current thread regarding being stuck on ‘Loading Data’.