ready all they need to do on Godady is log in and change the DNS setting which will instantly redirect all traffic to the server which the new website is stored on.
I’ve done this exact identical step on Godaddy myself when having a website built for me that was stored on the developer’s server and never left that server.
I then optimised it myself and appeared in the top ten search results on Google and Yahoo for the whole time that the website was live.