I did it some months ago. Was very smooth, no effort needed. Though I don't have any experience with the Docker install. I suppose you just change the image name and be done. Maybe take a backup of your database and data volumes to be safe. Here's the conversation from back then, but that wasn't about docker either, but includes quite some discussion of continuwuity vs tuwunel:

https://awful.systems/post/5029223

Edit: They have some sentences about the migration on their releases page: https://forgejo.ellis.link/continuwuation/continuwuity/releases

I upgraded to tuwunel (a different fork of conduwuit), and the process was advertised: just update the docker image. It's unlikely, but it's possible that since forking there have been breaking changes and would require some effort. But you'd have to check their release notes

Hey, I'm glad you got it working, yeah Continuwuity is still a drop-in replacement right now so all is good. I also did a drop-in a few months back with zero hassle at all. Most env vars and config options weren't changed, but I redid my continuwuity.toml file just to be sure.

I'd recommend joining the announcement room to keep track of updates, and the community room for support too