Manually uploading activities from Garmin Fit app
I had two activities recorded on 12 Nov 2015 that weren’t syncing to Garmin Connect from the Garmin Fit app on my phone. I signed out and back in a number of times to try to force the sync (as per Garmin's support page), but to no avail.
I eventually used Airdroid to download the two .fit files (stored in /SD Card/Android/data/com.garmin.android.apps.fitness/files/) from my phone to my laptop.
I was able to manually import those files into Garmin Connect and can now see the two activities as expected.
After signing out and back in to the Garmin Fit app I noticed that the 2 activities on 12 Nov 2015 were now duplicated - presumably the versions on the phone were slightly different from the versions on the website. I fixed this by removing all the Garmin Fit app data from my phone and signing in one final time.
Garmin Fit app has been removed
While writing this I tried to find a link to the Garmin Fit app in the Google Play Store. While I can see it in my list of installed apps, it no longer appears in the public store. I also followed the link to the app in iTunes but that reports that it's no longer available in the UK store. I can't find any official information about the app being removed but this thread on the Garmin forums suggests that it was removed in early 2015!.