Hi @benra,
After clicking 'Publish this version', you won't see a pop-up or success message - the app will just start publishing in the background and might take a short period to go into effect. A way to check if your app published app version has changed, is by:
- Go to make.powerapps.com
- Go the app section (left navigation panel)
- Select the three dots next to your app and click 'Details'
- Go to version history
This screen allows you to check what version is live.

As said, it might take some time to fully publish. If you don't see the newest version you could:
- Reopen the app and refresh your browser screen
- Close your browser & try to open the app again
- Clear your browser cache
It might be beneficial to check whether this issue persists when using another browser.
If this solves your question, would you be so kind as to accept it as a solution.
Thanks!