To summarise what's above (and to perhaps add a little to it):
1. Nobody has the right to see the will of a living person (or even to know that it exists although, obviously, it helps if someone can find it easily when the time comes).
2. In the UK the will of a deceased person becomes a public document, accessible to anyone, once probate has been granted. (It's actually the grant of probate which is the public document but the will is attached).
3. Until probate has been granted, nobody other than the executors has the right to view the will. (Obviously though, they might need to show the will to certain organisations, such as the deceased person's bank, in order to be able to collect the information required for the probate process to be completed).
4. If probate has been granted in England or Wales, New Judge has provided the relevant link above.
5. If probate was granted in Scotland more than 10 years ago then it's accessible via the National Archives of Scotland. If probate was granted in Scotland within the past 10 years you need to apply to the Edinburgh Sheriff Court.
6. If probate was granted in Northern Ireland you need to contact PRONI:
https://www.nidirect.gov.uk/proni