Thank you for asking and feedback.
Automatically, not. You’d have to change the URL structure or links either manually, or using some script if they’re in your database 
Otherwise, temporary you can rewrite URLs or redirect at Cloudflare dashboard using Redirect Rules or Transform Rules.
Might depend a bit which choice should you pick what kind of type of application you’re using
Only the image URL.
If they were pointed to https://www.yourdomain.com/images/filename.jpg, now after you’ve uploaded them to your R2 bucket and connected sub-domain such as storage.yourdomain.com, they should pointed to the https://storage.yourdomain.com/images/filename.jpg.
Sounds familiar to me, may I ask if it is a WordPress related URL structure?
You can keep it “as-is”, and using a simple code to rewrite this at your web server, without touching or changing anything.
See recent topic about this, and what kind of issue you might get when you’d be using a plugin to upload-sync with R2 bucket, if related to your case as well due to the caching and not showing “thumbnail” image on WordPress Media Library upon upload: