DNS caching is when a website's DNS settings are stored on your computer and ISP (Internet service provider) to speed up web browsing and reduce traffic. This means that if people are viewing a webpage from different providers they may see your old site and your new site at various times after the domain settings are first changed.
ISPs update their website caches at varying lengths of time. Some ISP's adhere to the TTL (time to live) settings in your DNS settings in your domain control panel, while others update their records every 2-3 days. Therefore, different devices or people using different networks may see your old website for some time after the DNS settings have been changed and your website is live.
This is DNS caching. Unfortunately, it cannot be sped up so if you're still seeing your old site and all your DNS records are correct you'll need to just keep an eye on your domain for a day or two to check that the new website and content is showing.
If you are announcing your new website launch via email or social media you may want to wait a day or two to allow time for your new site to appear for all visitors.
Generally, Name Server changes take up to 24 hours to change, while DNS settings take around 1 hour to change.