When DNS changes are made, they are not instantaneously visible across the Internet at large.
As DNS is a heavily cached, distributed system with no central location where all DNS information is stored, any DNS change takes time to reach other parts of the system. This period of time is often referred to as DNS propagation.
When you change a domain's authoritative name servers, you have no control over how long it takes the new authoritative name servers to propagate. Changes to your domain's authoritative name servers can take anywhere from one hour to an entire day depending on many factors, including your registrar and which top-level domain (for example, ".com" or ".org") your domain uses. During that period of time, some DNS resolvers will be using the old name servers while others are using the new name servers.
For changes to individual DNS records, the propagation time is partly dependent on your DNS zone's "time to live" (TTL) value, which tells others how long to cache responses. Typically, these values range from 5 minutes to 48 hours. However, many DNS resolvers ignore the TTL specified in your DNS zone and will cache records longer than you indicate.
Unfortunately, you cannot do anything to speed up DNS propagation once you have changed a DNS record. Lowering a DNS record's TTL only helps if you lower the TTL at least one day before you change the DNS record.