I couldn't find the chords for this song anywhere so I thought I'd take a stab at it. 
I'm pretty sure the song uses the same four chords over
and over. If you listen to the piano in the intro you can hear the third chord is a 
variation (I think with a 7) and I'm unsure about the second
and fourth chords, but this sounds pretty good, even playing along with the track. 
Comments welcome.

Standard tuning, Capo 1

Intro: G Cadd9 D Cadd9

G            Cadd9 D                 Cadd9
It's funny how the smile in your eyes
G          Cadd9              D            Cadd9
Is slowly fading with the last days in July
G              Cadd9          D                  Cadd9
The year is drowning like the calendar grows thin
G                Cadd9              D                 Cadd9
You'll sail away again with your back against the wind


Chorus:

            G          Cadd9             D Cadd9... sim.
And when the cold wind blows, you're gone
And I will hold my head again
But you can't run from the truth
Of oooh the London rain

Please send me beaches when my world is cold and weak
And press me flowers in the pages of the diaries you keep
It still won't fill the spot beside me in the bed where you sleep
As the calendar repeats

And when the cold wind blows, you're gone
And I will hold my head again
But you can't run from the truth
Of oooh the London rain

Yeah, just say it like you mean it
A last look of summer on your face
I swear I believe that
That someday you will stay
I can't take or leave it
I'll take it anywhere with you
I'll take it anywhere

And when the cold wind blows, you're gone
And I will hold my head again
But you can't run from the truth
Of oooh the London rain
[ Lyrics from: http://www.lyricsmode.com/lyrics/w/we_the_living/london_rain.html ]