When you change the nameservers of a domain, it takes some time to update the nameservers. This period is called DNS Propagation. During propagation, you may notice some abnormal activity of your domain/website like one person may see the new server while someone else sees the old one, yourdomain.com may work while www.yourdomain.com may not work, etc. However, these abnormal activities are actually normal during the DNS Propagation.
Usually, DNS Propagation takes up to 24 hours. But depending on various factors, it may take a maximum of 72 hours. So if you are getting these types of issues within this time, there is nothing to worry actually. But everything must be fine within this period. You can check your DNS Status anytime using DNS Checker. If you are facing any issue after 72 hours, then there is something wrong. So you should contact your hosting service provider.
In short, DNS propagation depends upon how frequently the DNS master cache nameservers are refreshed.