The government has responded to the petition requesting that the English national Anthem be changed to Jerusalem, by stating that they will not do it. You can read the response here, but it is copied below
The national anthem of the UK ‘God Save the Queen’ only has that status by convention. It has no official status either by legislation or royal proclamation. Similarly, ‘Flower of Scotland’, although it is regarded in some quarters as the Scottish national anthem, has no official status. Insofar as a anthem for England is concerned, ‘Land of Hope and Glory’ is often played when England play in sporting competitions (such as the Commonwealth Games) against other countries of the UK. However, this too has no official standing. The Government has no plans to legislate on this issue.
So it appears that the United Kingdom has no official anthem.