How do I update the URL after changing the hosting of a website?
There are a couple of possibilities when this happens. The first is to make sure that you still have an .htaccess file, as moving between hosts sometimes doesn't include the hidden files. 

If you do have it, depending on the server, the Symlinks may need to be changed. For instance my Ubuntu server needs the .htaccess file to be changed to this in order for it to work

#Options +FollowSymLinks // Commented out
Options +SymLinksIfOwnerMatch


The next possibility is that the DNS propagation hasn't finished yet. It can take up to 72 hours with some hosts before everything is fully refreshed and pointing the right location.