A few hours ago I got a message to do something or other, which I did, and all of a sudden the app is working again. (IIRC it was setting permissions i.e. continuing on from where it gave up the ghost the last time I tried.)
I checked and the last time the app on the laptop was modified was 22 August (eight days ago) and on the phone the last update was 1 August, so I've no idea what's going on.
Very strange.
I gather from the comments here this has happened to some other people but it didn't last. I'll see if it sticks. Hopefully it will. If it doesn't I'll just go back to Pushbullet, which works fine.
Well, the program works fine for me and the whole process appears stable.