In reply to BusyLizzie:
For the sake of Strava's good name I should confess the ending to this saga.
A few weeks ago (I now remember) I found the "battery saver mode" thingy on my phone, thought "That's a good idea" and switched it on then forgot I'd done so.
It was at about the same time (I now notice) that Strava stopped working. After trying lots of ideas I tried a new app - Runkeeper, which is free and seems very good. Tried using it last week and it said something like "nope, can't do GPS, you've got battery save mode on." Oh ho, I thought, so I have, and turned it off.
And now Strava works perfectly again.
Silly me...