I've always favoured "Yellow Submarine" as a replacement to God Save the Queen.
Good natured, inclusive, and absurd, which are pretty good candidates for what ought to be seen as the "British Values" worth encouraging.

Paul McCartney's Frogs' Chorus wouuld be another excellent choice -

Win Or Lose, Sink Or Swim
One Thing Is Certain We'll Never Give In
Side By Side, Hand In Hand
We All Stand Together...