1040 only uploads to Strava over wifi

I'm pretty sure the Edge units don't upload to any other app than Garmin Connect.  It's Garmin Connect that subsequently uploads activities to apps that you have set up. First it must upload to GC, then it should propagate to the connected apps.

So, just to clarify, my phone syncs with Garmin Connect over Bluetooth, but my rides only appear in Strava once I am on WiFi. 

In that case, if my understaning is correct, then the issue is not with your 1040 or, almost certainly, with Connect. As looigi has pointed out, your activities sync server-to-server, Connect to Strava (and / or others), and how you connect to the internet is irrelevant to this process *once your activity has been uploaded to Connect* (that's not to say that the sync process doesn't sometimes fail, but it's not caused by your connection). 

The only thing I can think of is that your 4/5G connection is poor, and that until you connect to wifi Strava is unable to refresh. I'm a bit surprised by that, though; it's not something I consider a lot, but I'd tend to guess that the upload process to Connect would require better data transfer than refreshing the Strava activity feed. 

I’ve seen these sorts of issues a number of times with both 1030 and 1040. I find usually a full power off and on again fixes it