After the content transfer, you can create an A record in your existing DNS (most likely on your old server) to redirect all visitors to the new server; this is much faster than waiting for the propagation of new name servers.
Also, after the site transfer, you can create a MX record so your emails will all be delivered to the new server and not need to wait as long for propagation.
Once you have the A records and MX records updated to point to the new server, you can change the name servers anytime.