The reason why it takes time the first time you set it up, has nothing to do with Anvil. The DNS servers around the world need to know where to go when your domain is looked up, and it takes time for the whole world to know about you.
After that has been setup, all happens in Anvil. When one client looks for that domain, Anvil serves your app, which is immediately available after each editing.
If you edit your app and don’t see the changes in the app running on the custom domain, perhaps you need to publish if you are working on the old editor or you need to merge from the development to the published environment if you work on the new editor.
Tony did not answer your question, he just gave you a general advice. A good one, but it has nothing to do with your question.