Does WordPress automatically detect the delegating of a web hosting account?

Recently I have been working with WordPress on the web
hosting account before delegating it in my client’s web hosting on their domain name. So could someone please tell me, if WordPress will automatically detect, once I have delegated the web hosting account with the domain name? Or do I have to change the WordPress address (URL) and Site Address (URL) once delegated? The domain name I have been working before delegating is like this: https://res04.webhostingsupplier.com/~myclient, and then when I delegate it, it should look something like this: myclient.com. Thanks in advance.