For whatever reason, when I select "continue" I get a "Game updated!" line on the top of the screen, and the game never seems to progress past the loading screen. What could be causing this?

Same thing happens to me. No answer found yet.

I contacted the devs. I have set the exe to "run as an administrator" (I am not sure if it's essential) and deleted the two folders in Appdata/local named "Bilkins Folly" and "BilkinsFolly". The game recreated the folders once it saw they weren't there, and I was able to save and load normally afterward.

I lost the progress I made before this (although it's more accurate to say that the game didn't save before), but I think that's unavoidable.

Edit: I wish I could mark my own replies as the solution. XD

Yes, it appears that if you turn off cloud sync in Gog Galaxy options and then let the game save for the very first time, you can then turn it back on and it will continue to work fine.

I suggest to fix this, it is really unintuitive to use this way. Cloud sync is by default on for most of us.

I encountered this exact issue and this worked for me, thank you!