If the uninhabited island is within the UK (e.g. in the middle of the Thames), you'd need permission from the legal owner of the land together with planning permission from the local authority. In practice, you'd almost certainly be denied permission by the planning authorities.
If the island is outside of the UK, you'd need permission from the government which claimed the island as part of its territory (and also from the land owner, if this wasn't the relevant government). Government's tend to disapprove of the 'invasion' of their territory. For instance, a group of scrap metal merchants raising a flag on the uninhabited island of South Georgia was enough to trigger the Falklands War!
Governments won't normally give approval for anyone to build on their uninhabited islands. (Apart from the political implications, many small islands are unique nature habitats, which should be left undisturbed by man). Sometimes they'll approve the construction of temporary buildings for scientific research teams. Alternatively, they might be tempted to take the money from a billionaire who offered them a vast amount to have the island made available for his/her exclusive use. However, the chances of you or me being allowed to build on such an island are effectively nil.
If you're determined to find some isolation, you could try finding an uninhabited man-made installation, in international waters and declaring it to be an independent state:
http://en.wikipedia.org/wiki/Principality_of_S ealand
Chris