Well, first you've got to go through the provincial anthems:
Scotland the Brave
Land of Our Fathers
What Shall We Do With the Drunken Sailor
When the Boat Comes In
Anything from the Mother Goose collection
Early One Morning
Down by the Sally Gardens
Maggie Mae (not the Rod Stewart song of the same name)
Ilkley Moor Bah'tat
And Shall Trelawney Die
Charlie is my Darling
English Country Garden
Down at the Old Bull and Bush
Counting the Goats
There's a few to be getting on with. There are plenty more. Don't forget too that England is not the only place that is British. Irish songs were considered part of the same heritage until the last century, but Scottish songs may cease to be considered British if they get their independence later in this decade. Wales was called The Land of Song, and have a long list of hymns sung as folk songs.
You need to think about the period. Greensleeves was written by King Henry VIII in the 16th century, but Land of Hope and Glory (the music written by Elgar in my home town to Benson's words) is less than a century old.
Other foreign songs have been naturalised. The Halleluiah Chorus, widely sung throughout Britain was written in England by an 18th century German composer. You'll Never Walk Alone, the famous Liverpool football anthem is actually an American song from Carousel, a 20th century musical.
Lastly, if you want to be up-to-date with stereotypical British singing, we must not forget the drinking song 'Ere We Go, heard up and down the land in British cities at chucking out time on a Saturday night, although these days the words are a bit slurred and confused by vomit and is barely recognisable as a song.