Domain registrars, such as 123-reg, advise that it can take between 24 and 48 hours for DNS settings to fully finish updating (or propagating).
While it can take this length of time for all browsers and devices to see the new website, you will normally be able to see the changes on some devices not long after the change is made. Mobile phones tend to update faster than desktops, so you can check the new site is running on your phone first. Your desktop and that of your regular visitors will take a longer to update than a new visitor because frequent visitors will have the old website saved in their browser memory (or cache). If you see the new website on your mobile or another device, you can rest assured that the new site is set up correctly and will show on all devices within the 48 hour period.
If a nameserver is changed or if the domain is being transferred, it will take longer for any device even a mobile to start seeing the new website.