What's the name of that Popular British instrumental in the movies?

I think that the song you are referring to is the chorus of the song 'Rule Britannia' which was composed by Thomas Arne in 1740 from the words of James Thompson's patriotic poem.

Here it is transposed onto your very helpful 'DAs'

Daaa Da Da Da, Da daaaaaaah Da da da daaaaaaaah!

Rule Brit-ann-ia, Brit - ann - ia rules the waves!

It is always played as a result of some very unimaginative audio choices when can't think of any other song (especially on which is about London, rather than about Britain as a whole) which reminds people of Britain.

It's always funny to see Britain in American films (Speed is a very good example of this) where, to emphasise that we're in Britain now, we get 'London - England' at the bottom of the screen (as opposed to 'London - Wisconsin') followed quickly by a shot of the houses of Parliament, pouring with rain, with a red double decker bus on Westminster bridge in the foreground being followed by a black cab from which a man with a suit, bowler hat and umbrella steps out. The chimes of Big Ben can be heard softly in the background.

That doesn't actually help very much.

If you could find a movie where they're actually playing this, it would be a lot easier to work it out, as we could listen to it ourselves.

My guess is that it might be something by Elgar, Vaughan-Williams, or Walton (all late 19th/20th century British classical composers), but I really can't work it out without hearing it.