Although it is officially the British national anthem, there is some feeling of vacillation
about whether it is more British, English, or royal
in character. At sport
ing events where England, Scotland, Wales, and Northern Ireland compete as separate countries it is inappropriate to use it as the England song.
Also it was the national anthem of Australia (and possibly other Commonwealth countries, I don't know) until fairly recently; supporters of its retention argued it was a royal anthem to the Australian head of state, the Queen, while opponents argued it was essentially the anthem of another country, Britain.
The tune is also used for the national anthem of Liechtenstein, Oben am jungen Rhein, "High up on the young Rhine"; and for the American song My Country 'Tis of Thee.
As to its being "official", it was changed in the Proclamation of the present Queen on 7 February 1952. The King died on 6 February while Princess Elizabeth was in Kenya. On her arrival home, she was "proclaimed" the following day in the City of London. (I don't know what "proclaim" technically means here, since the Crown passes to the next in line at the instant of death, and the Coronation is about a year later.)
I once looked back in Keesing's, the weekly (as it then was) archive of world events, and read their extremely detailed account of the events surrounding the King's death and the Queen's accession. The Proclamation dealt with may things, but among its technical details, the national anthem was changed from "God Save the King" to "God Save the Queen", the courts of the King's Bench were changed to Queen's Bench, and the lawyers called King's Counsels were changed to Queen's Counsels.
Most countries are much simpler: Chad or Togo became independent on such-and-such a date in 1960, with a brand-new constitution that stated inter alia that the national anthem is such-and-such, the flag is such-and-such. Britain's constitution has grown organically (with some pruning and forcing) for many centuries, and probably there was no one date on which an official declaration adopted the national anthem. (See my article on the Union Jack for an account of its complex history, as an example.)