Thank you Garmin for a wasted hour. After foolishly agreeing to a software update (6.9.1.0 to 6.10.0.0, I think) when prompted by Express, my Forerunner 610 would no longer sync. Got it going again eventually after this sequence:
Restart computer (Windows 10) - no change
Uninstall Garmin Express
Re-install Garmin Express 6.9.1.0 from download page - still no joy
Remove device from Express
Add device back to Express - took ages setting up, but finally solved the problem and was able to sync activity.
Maybe if I had just removed and re-added the device it would have worked, who knows?
I will not be updating to 6.10.0.0 again in a hurry.