Whenever you make changes to your domain's nameservers, they need to reflect across the internet. This process, which usually takes anywhere between 24-48 hours to complete, is called DNS Propagation.
Why do you have to wait for this long?
: All Internet Service Providers (ISPs) across the globe store a copy of your domains DNS whenever your website is resolved through their network. They do this, so that your website loads faster when someone visits it the next time. However, when you change your domains DNS or name servers, this copy or cache is not cleared immediately and has to wait for the ISP's to clear their cache and store a fresh copy of your domains updated DNS information.
Can I speed this up?
: There is nothing you can do to accelerate the propagation. However, you can plan for propagation to minimize the downtime affects.
Tip for Minimising Downtime - Once you have uploaded your web content on the new server, you can add DNS records in your existing DNS zone (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. Once you have the DNS records updated to point to the new server, you can change the name servers anytime.
How do I know when propagation is complete?
: Unfortunately there are no tools available to track name server propagation. Even a Whois check, ping, and traceroute test cannot indicate whether the propagation is complete. Certain tools similar to WhatsMyDNS give you some information regarding name server propagation. However, it may not show you accurate and complete information.
When we make any DNS changes, we sit back and relax. The changes gets reflected in a day or two (maximum of 48 hours). We'd suggest you to do the same.