Using the export functionality for my activity:
https://connect.garmin.com/modern/activity/799310730
I've then tried to load the resulting TCX or GPX file into another bit of software but I am getting errors saying the file contains no tracks or missing time information etc.
Since the run clearly has data (I can see the pace, route etc on Garmin Connect) there must be a bug in the exporting that's messing things up.
It's worth noting that it is only affecting this activity - other activities export perfectly.
SOLUTION:
Thanks to Grendelino the "fix" is to export to a FIT file (Using "Export Original") and then upload and/or convert it with other software to a GPX.